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3.1834 \(\int \frac {A+B x}{(d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=435 \[ -\frac {3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac {3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt {d+e x} (b d-a e)^8}+\frac {1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac {3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac {429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac {143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac {13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

[Out]

3003/640*e^4*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^6/(e*x+d)^(5/2)+1/5*(-A*b+B*a)/b/(-a*e+b*d)/(b*x+a)^5/(e*x+
d)^(5/2)+1/8*(3*A*b*e-B*a*e-2*B*b*d)/b/(-a*e+b*d)^2/(b*x+a)^4/(e*x+d)^(5/2)+13/48*e*(-3*A*b*e+B*a*e+2*B*b*d)/b
/(-a*e+b*d)^3/(b*x+a)^3/(e*x+d)^(5/2)-143/192*e^2*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^4/(b*x+a)^2/(e*x+d)^(5
/2)+429/128*e^3*(-3*A*b*e+B*a*e+2*B*b*d)/b/(-a*e+b*d)^5/(b*x+a)/(e*x+d)^(5/2)+1001/128*e^4*(-3*A*b*e+B*a*e+2*B
*b*d)/(-a*e+b*d)^7/(e*x+d)^(3/2)-3003/128*b^(3/2)*e^4*(-3*A*b*e+B*a*e+2*B*b*d)*arctanh(b^(1/2)*(e*x+d)^(1/2)/(
-a*e+b*d)^(1/2))/(-a*e+b*d)^(17/2)+3003/128*b*e^4*(-3*A*b*e+B*a*e+2*B*b*d)/(-a*e+b*d)^8/(e*x+d)^(1/2)

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Rubi [A]  time = 0.49, antiderivative size = 435, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {27, 78, 51, 63, 208} \[ -\frac {3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac {3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt {d+e x} (b d-a e)^8}+\frac {1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac {3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac {429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac {143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac {13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac {a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac {A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (A*b - a*B)/(5*b*(b*d - a*e)*(a
 + b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e + a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13
*e*(2*b*B*d - 3*A*b*e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d - 3*A*b*e
 + a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b
*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) + (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(
3/2)) + (3003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003*b^(3/2)*e^4*(2*b*B*
d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(17/2))

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
 n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {A+B x}{(a+b x)^6 (d+e x)^{7/2}} \, dx\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}+\frac {(2 b B d-3 A b e+a B e) \int \frac {1}{(a+b x)^5 (d+e x)^{7/2}} \, dx}{2 b (b d-a e)}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}-\frac {(13 e (2 b B d-3 A b e+a B e)) \int \frac {1}{(a+b x)^4 (d+e x)^{7/2}} \, dx}{16 b (b d-a e)^2}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}+\frac {\left (143 e^2 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{7/2}} \, dx}{96 b (b d-a e)^3}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}-\frac {\left (429 e^3 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{7/2}} \, dx}{128 b (b d-a e)^4}\\ &=-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{7/2}} \, dx}{256 b (b d-a e)^5}\\ &=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {\left (3003 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{5/2}} \, dx}{256 (b d-a e)^6}\\ &=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {\left (3003 b e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^7}\\ &=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^4 (2 b B d-3 A b e+a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^8}\\ &=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}+\frac {\left (3003 b^2 e^3 (2 b B d-3 A b e+a B e)\right ) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 (b d-a e)^8}\\ &=\frac {3003 e^4 (2 b B d-3 A b e+a B e)}{640 b (b d-a e)^6 (d+e x)^{5/2}}-\frac {A b-a B}{5 b (b d-a e) (a+b x)^5 (d+e x)^{5/2}}-\frac {2 b B d-3 A b e+a B e}{8 b (b d-a e)^2 (a+b x)^4 (d+e x)^{5/2}}+\frac {13 e (2 b B d-3 A b e+a B e)}{48 b (b d-a e)^3 (a+b x)^3 (d+e x)^{5/2}}-\frac {143 e^2 (2 b B d-3 A b e+a B e)}{192 b (b d-a e)^4 (a+b x)^2 (d+e x)^{5/2}}+\frac {429 e^3 (2 b B d-3 A b e+a B e)}{128 b (b d-a e)^5 (a+b x) (d+e x)^{5/2}}+\frac {1001 e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^7 (d+e x)^{3/2}}+\frac {3003 b e^4 (2 b B d-3 A b e+a B e)}{128 (b d-a e)^8 \sqrt {d+e x}}-\frac {3003 b^{3/2} e^4 (2 b B d-3 A b e+a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{17/2}}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 100, normalized size = 0.23 \[ \frac {\frac {5 a B-5 A b}{(a+b x)^5}-\frac {5 e^4 (-a B e+3 A b e-2 b B d) \, _2F_1\left (-\frac {5}{2},5;-\frac {3}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}}{25 b (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

((-5*A*b + 5*a*B)/(a + b*x)^5 - (5*e^4*(-2*b*B*d + 3*A*b*e - a*B*e)*Hypergeometric2F1[-5/2, 5, -3/2, (b*(d + e
*x))/(b*d - a*e)])/(b*d - a*e)^5)/(25*b*(b*d - a*e)*(d + e*x)^(5/2))

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fricas [B]  time = 1.20, size = 6033, normalized size = 13.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(45045*(2*B*a^5*b^2*d^4*e^4 + (B*a^6*b - 3*A*a^5*b^2)*d^3*e^5 + (2*B*b^7*d*e^7 + (B*a*b^6 - 3*A*b^7)*
e^8)*x^8 + (6*B*b^7*d^2*e^6 + (13*B*a*b^6 - 9*A*b^7)*d*e^7 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^8)*x^7 + (6*B*b^7*d^3
*e^5 + 3*(11*B*a*b^6 - 3*A*b^7)*d^2*e^6 + 5*(7*B*a^2*b^5 - 9*A*a*b^6)*d*e^7 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e^8
)*x^6 + (2*B*b^7*d^4*e^4 + (31*B*a*b^6 - 3*A*b^7)*d^3*e^5 + 15*(5*B*a^2*b^5 - 3*A*a*b^6)*d^2*e^6 + 10*(5*B*a^3
*b^4 - 9*A*a^2*b^5)*d*e^7 + 10*(B*a^4*b^3 - 3*A*a^3*b^4)*e^8)*x^5 + 5*(2*B*a*b^6*d^4*e^4 + (13*B*a^2*b^5 - 3*A
*a*b^6)*d^3*e^5 + 18*(B*a^3*b^4 - A*a^2*b^5)*d^2*e^6 + 2*(4*B*a^4*b^3 - 9*A*a^3*b^4)*d*e^7 + (B*a^5*b^2 - 3*A*
a^4*b^3)*e^8)*x^4 + (20*B*a^2*b^5*d^4*e^4 + 10*(7*B*a^3*b^4 - 3*A*a^2*b^5)*d^3*e^5 + 30*(2*B*a^4*b^3 - 3*A*a^3
*b^4)*d^2*e^6 + (17*B*a^5*b^2 - 45*A*a^4*b^3)*d*e^7 + (B*a^6*b - 3*A*a^5*b^2)*e^8)*x^3 + (20*B*a^3*b^4*d^4*e^4
 + 10*(4*B*a^4*b^3 - 3*A*a^3*b^4)*d^3*e^5 + 3*(7*B*a^5*b^2 - 15*A*a^4*b^3)*d^2*e^6 + 3*(B*a^6*b - 3*A*a^5*b^2)
*d*e^7)*x^2 + (10*B*a^4*b^3*d^4*e^4 + (11*B*a^5*b^2 - 15*A*a^4*b^3)*d^3*e^5 + 3*(B*a^6*b - 3*A*a^5*b^2)*d^2*e^
6)*x)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sqrt(b/(b*d - a*e)))/(b*x + a
)) + 2*(768*A*a^7*e^7 + 96*(B*a*b^6 + 4*A*b^7)*d^7 - 16*(62*B*a^2*b^5 + 213*A*a*b^6)*d^6*e + 28*(187*B*a^3*b^4
 + 498*A*a^2*b^5)*d^5*e^2 - 70*(332*B*a^4*b^3 + 519*A*a^3*b^4)*d^4*e^3 - (100363*B*a^5*b^2 - 79905*A*a^4*b^3)*
d^3*e^4 - 1024*(16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 + 512*(B*a^7 - 18*A*a^6*b)*d*e^6 - 45045*(2*B*b^7*d*e^6 + (
B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 105105*(2*B*b^7*d^2*e^5 + (5*B*a*b^6 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6
)*e^7)*x^6 - 3003*(46*B*b^7*d^3*e^4 + 3*(117*B*a*b^6 - 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^
6 + 128*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 2145*(6*B*b^7*d^4*e^3 + (307*B*a*b^6 - 9*A*b^7)*d^3*e^4 + 12*(83*
B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 - 211*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)
*x^4 + 715*(4*B*b^7*d^5*e^2 - 2*(43*B*a*b^6 + 3*A*b^7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e^4 - 2*(1
547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a^4*b^3 - 1686*A*a^3*b^4)*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*
b^3)*e^7)*x^3 - 65*(16*B*b^7*d^6*e - 12*(17*B*a*b^6 + 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^
3 + 2*(8837*B*a^3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*e^5 + 3*(1867*B*a^5*
b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5*b^2)*e^7)*x^2 + 5*(96*B*b^7*d^7 - 16*(59*B*a*b^6 + 9*A*b^
7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*a*b^6)*d^5*e^2 - 42*(491*B*a^3*b^4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*
a^4*b^3 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(20687*B*a^5*b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 1536*(5*B*a^6*b - 16*A*a^
5*b^2)*d*e^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)*sqrt(e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^
9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7
 + a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 + 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^
4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*
a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^
6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*
e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 350*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*
e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a
*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6
*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b
^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e - 14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7
*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^
9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a
^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^
10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*
e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6
 + 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a
^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10
*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e^8 + 3*a^13*d^2*e^9)*x), -1/1920*(45045*(2*B*a^5*b^2*d^4*e
^4 + (B*a^6*b - 3*A*a^5*b^2)*d^3*e^5 + (2*B*b^7*d*e^7 + (B*a*b^6 - 3*A*b^7)*e^8)*x^8 + (6*B*b^7*d^2*e^6 + (13*
B*a*b^6 - 9*A*b^7)*d*e^7 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^8)*x^7 + (6*B*b^7*d^3*e^5 + 3*(11*B*a*b^6 - 3*A*b^7)*d^
2*e^6 + 5*(7*B*a^2*b^5 - 9*A*a*b^6)*d*e^7 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e^8)*x^6 + (2*B*b^7*d^4*e^4 + (31*B*a
*b^6 - 3*A*b^7)*d^3*e^5 + 15*(5*B*a^2*b^5 - 3*A*a*b^6)*d^2*e^6 + 10*(5*B*a^3*b^4 - 9*A*a^2*b^5)*d*e^7 + 10*(B*
a^4*b^3 - 3*A*a^3*b^4)*e^8)*x^5 + 5*(2*B*a*b^6*d^4*e^4 + (13*B*a^2*b^5 - 3*A*a*b^6)*d^3*e^5 + 18*(B*a^3*b^4 -
A*a^2*b^5)*d^2*e^6 + 2*(4*B*a^4*b^3 - 9*A*a^3*b^4)*d*e^7 + (B*a^5*b^2 - 3*A*a^4*b^3)*e^8)*x^4 + (20*B*a^2*b^5*
d^4*e^4 + 10*(7*B*a^3*b^4 - 3*A*a^2*b^5)*d^3*e^5 + 30*(2*B*a^4*b^3 - 3*A*a^3*b^4)*d^2*e^6 + (17*B*a^5*b^2 - 45
*A*a^4*b^3)*d*e^7 + (B*a^6*b - 3*A*a^5*b^2)*e^8)*x^3 + (20*B*a^3*b^4*d^4*e^4 + 10*(4*B*a^4*b^3 - 3*A*a^3*b^4)*
d^3*e^5 + 3*(7*B*a^5*b^2 - 15*A*a^4*b^3)*d^2*e^6 + 3*(B*a^6*b - 3*A*a^5*b^2)*d*e^7)*x^2 + (10*B*a^4*b^3*d^4*e^
4 + (11*B*a^5*b^2 - 15*A*a^4*b^3)*d^3*e^5 + 3*(B*a^6*b - 3*A*a^5*b^2)*d^2*e^6)*x)*sqrt(-b/(b*d - a*e))*arctan(
-(b*d - a*e)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))/(b*e*x + b*d)) + (768*A*a^7*e^7 + 96*(B*a*b^6 + 4*A*b^7)*d^7 -
 16*(62*B*a^2*b^5 + 213*A*a*b^6)*d^6*e + 28*(187*B*a^3*b^4 + 498*A*a^2*b^5)*d^5*e^2 - 70*(332*B*a^4*b^3 + 519*
A*a^3*b^4)*d^4*e^3 - (100363*B*a^5*b^2 - 79905*A*a^4*b^3)*d^3*e^4 - 1024*(16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 +
 512*(B*a^7 - 18*A*a^6*b)*d*e^6 - 45045*(2*B*b^7*d*e^6 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 105105*(2*B*b^7*d^2*e^
5 + (5*B*a*b^6 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 - 3003*(46*B*b^7*d^3*e^4 + 3*(117*B*a*b^6
 - 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^6 + 128*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 2145*(6
*B*b^7*d^4*e^3 + (307*B*a*b^6 - 9*A*b^7)*d^3*e^4 + 12*(83*B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 -
 211*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 715*(4*B*b^7*d^5*e^2 - 2*(43*B*a*b^6 + 3*A*b^
7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e^4 - 2*(1547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a
^4*b^3 - 1686*A*a^3*b^4)*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 - 65*(16*B*b^7*d^6*e - 12*(17*B*a*b^6
+ 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^3 + 2*(8837*B*a^3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*
(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*e^5 + 3*(1867*B*a^5*b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5
*b^2)*e^7)*x^2 + 5*(96*B*b^7*d^7 - 16*(59*B*a*b^6 + 9*A*b^7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*a*b^6)*d^5*e^2
- 42*(491*B*a^3*b^4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*a^4*b^3 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(20687*B*a^5*
b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 1536*(5*B*a^6*b - 16*A*a^5*b^2)*d*e^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)*sqrt(
e*x + d))/(a^5*b^8*d^11 - 8*a^6*b^7*d^10*e + 28*a^7*b^6*d^9*e^2 - 56*a^8*b^5*d^8*e^3 + 70*a^9*b^4*d^7*e^4 - 56
*a^10*b^3*d^6*e^5 + 28*a^11*b^2*d^5*e^6 - 8*a^12*b*d^4*e^7 + a^13*d^3*e^8 + (b^13*d^8*e^3 - 8*a*b^12*d^7*e^4 +
 28*a^2*b^11*d^6*e^5 - 56*a^3*b^10*d^5*e^6 + 70*a^4*b^9*d^4*e^7 - 56*a^5*b^8*d^3*e^8 + 28*a^6*b^7*d^2*e^9 - 8*
a^7*b^6*d*e^10 + a^8*b^5*e^11)*x^8 + (3*b^13*d^9*e^2 - 19*a*b^12*d^8*e^3 + 44*a^2*b^11*d^7*e^4 - 28*a^3*b^10*d
^6*e^5 - 70*a^4*b^9*d^5*e^6 + 182*a^5*b^8*d^4*e^7 - 196*a^6*b^7*d^3*e^8 + 116*a^7*b^6*d^2*e^9 - 37*a^8*b^5*d*e
^10 + 5*a^9*b^4*e^11)*x^7 + (3*b^13*d^10*e - 9*a*b^12*d^9*e^2 - 26*a^2*b^11*d^8*e^3 + 172*a^3*b^10*d^7*e^4 - 3
50*a^4*b^9*d^6*e^5 + 322*a^5*b^8*d^5*e^6 - 56*a^6*b^7*d^4*e^7 - 164*a^7*b^6*d^3*e^8 + 163*a^8*b^5*d^2*e^9 - 65
*a^9*b^4*d*e^10 + 10*a^10*b^3*e^11)*x^6 + (b^13*d^11 + 7*a*b^12*d^10*e - 62*a^2*b^11*d^9*e^2 + 134*a^3*b^10*d^
8*e^3 - 10*a^4*b^9*d^7*e^4 - 406*a^5*b^8*d^6*e^5 + 728*a^6*b^7*d^5*e^6 - 568*a^7*b^6*d^4*e^7 + 161*a^8*b^5*d^3
*e^8 + 55*a^9*b^4*d^2*e^9 - 50*a^10*b^3*d*e^10 + 10*a^11*b^2*e^11)*x^5 + 5*(a*b^12*d^11 - 2*a^2*b^11*d^10*e -
14*a^3*b^10*d^9*e^2 + 65*a^4*b^9*d^8*e^3 - 106*a^5*b^8*d^7*e^4 + 56*a^6*b^7*d^6*e^5 + 56*a^7*b^6*d^5*e^6 - 106
*a^8*b^5*d^4*e^7 + 65*a^9*b^4*d^3*e^8 - 14*a^10*b^3*d^2*e^9 - 2*a^11*b^2*d*e^10 + a^12*b*e^11)*x^4 + (10*a^2*b
^11*d^11 - 50*a^3*b^10*d^10*e + 55*a^4*b^9*d^9*e^2 + 161*a^5*b^8*d^8*e^3 - 568*a^6*b^7*d^7*e^4 + 728*a^7*b^6*d
^6*e^5 - 406*a^8*b^5*d^5*e^6 - 10*a^9*b^4*d^4*e^7 + 134*a^10*b^3*d^3*e^8 - 62*a^11*b^2*d^2*e^9 + 7*a^12*b*d*e^
10 + a^13*e^11)*x^3 + (10*a^3*b^10*d^11 - 65*a^4*b^9*d^10*e + 163*a^5*b^8*d^9*e^2 - 164*a^6*b^7*d^8*e^3 - 56*a
^7*b^6*d^7*e^4 + 322*a^8*b^5*d^6*e^5 - 350*a^9*b^4*d^5*e^6 + 172*a^10*b^3*d^4*e^7 - 26*a^11*b^2*d^3*e^8 - 9*a^
12*b*d^2*e^9 + 3*a^13*d*e^10)*x^2 + (5*a^4*b^9*d^11 - 37*a^5*b^8*d^10*e + 116*a^6*b^7*d^9*e^2 - 196*a^7*b^6*d^
8*e^3 + 182*a^8*b^5*d^7*e^4 - 70*a^9*b^4*d^6*e^5 - 28*a^10*b^3*d^5*e^6 + 44*a^11*b^2*d^4*e^7 - 19*a^12*b*d^3*e
^8 + 3*a^13*d^2*e^9)*x)]

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giac [B]  time = 0.44, size = 1676, normalized size = 3.85 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

3003/128*(2*B*b^3*d*e^4 + B*a*b^2*e^5 - 3*A*b^3*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^8*d^8 -
8*a*b^7*d^7*e + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2
*d^2*e^6 - 8*a^7*b*d*e^7 + a^8*e^8)*sqrt(-b^2*d + a*b*e)) + 1/1920*(90090*(x*e + d)^7*B*b^7*d*e^4 - 420420*(x*
e + d)^6*B*b^7*d^2*e^4 + 768768*(x*e + d)^5*B*b^7*d^3*e^4 - 677820*(x*e + d)^4*B*b^7*d^4*e^4 + 275990*(x*e + d
)^3*B*b^7*d^5*e^4 - 33280*(x*e + d)^2*B*b^7*d^6*e^4 - 2560*(x*e + d)*B*b^7*d^7*e^4 - 768*B*b^7*d^8*e^4 + 45045
*(x*e + d)^7*B*a*b^6*e^5 - 135135*(x*e + d)^7*A*b^7*e^5 + 210210*(x*e + d)^6*B*a*b^6*d*e^5 + 630630*(x*e + d)^
6*A*b^7*d*e^5 - 1153152*(x*e + d)^5*B*a*b^6*d^2*e^5 - 1153152*(x*e + d)^5*A*b^7*d^2*e^5 + 1694550*(x*e + d)^4*
B*a*b^6*d^3*e^5 + 1016730*(x*e + d)^4*A*b^7*d^3*e^5 - 965965*(x*e + d)^3*B*a*b^6*d^4*e^5 - 413985*(x*e + d)^3*
A*b^7*d^4*e^5 + 149760*(x*e + d)^2*B*a*b^6*d^5*e^5 + 49920*(x*e + d)^2*A*b^7*d^5*e^5 + 14080*(x*e + d)*B*a*b^6
*d^6*e^5 + 3840*(x*e + d)*A*b^7*d^6*e^5 + 5376*B*a*b^6*d^7*e^5 + 768*A*b^7*d^7*e^5 + 210210*(x*e + d)^6*B*a^2*
b^5*e^6 - 630630*(x*e + d)^6*A*a*b^6*e^6 + 2306304*(x*e + d)^5*A*a*b^6*d*e^6 - 1016730*(x*e + d)^4*B*a^2*b^5*d
^2*e^6 - 3050190*(x*e + d)^4*A*a*b^6*d^2*e^6 + 1103960*(x*e + d)^3*B*a^2*b^5*d^3*e^6 + 1655940*(x*e + d)^3*A*a
*b^6*d^3*e^6 - 249600*(x*e + d)^2*B*a^2*b^5*d^4*e^6 - 249600*(x*e + d)^2*A*a*b^6*d^4*e^6 - 30720*(x*e + d)*B*a
^2*b^5*d^5*e^6 - 23040*(x*e + d)*A*a*b^6*d^5*e^6 - 16128*B*a^2*b^5*d^6*e^6 - 5376*A*a*b^6*d^6*e^6 + 384384*(x*
e + d)^5*B*a^3*b^4*e^7 - 1153152*(x*e + d)^5*A*a^2*b^5*e^7 - 338910*(x*e + d)^4*B*a^3*b^4*d*e^7 + 3050190*(x*e
 + d)^4*A*a^2*b^5*d*e^7 - 275990*(x*e + d)^3*B*a^3*b^4*d^2*e^7 - 2483910*(x*e + d)^3*A*a^2*b^5*d^2*e^7 + 16640
0*(x*e + d)^2*B*a^3*b^4*d^3*e^7 + 499200*(x*e + d)^2*A*a^2*b^5*d^3*e^7 + 32000*(x*e + d)*B*a^3*b^4*d^4*e^7 + 5
7600*(x*e + d)*A*a^2*b^5*d^4*e^7 + 26880*B*a^3*b^4*d^5*e^7 + 16128*A*a^2*b^5*d^5*e^7 + 338910*(x*e + d)^4*B*a^
4*b^3*e^8 - 1016730*(x*e + d)^4*A*a^3*b^4*e^8 - 275990*(x*e + d)^3*B*a^4*b^3*d*e^8 + 1655940*(x*e + d)^3*A*a^3
*b^4*d*e^8 - 499200*(x*e + d)^2*A*a^3*b^4*d^2*e^8 - 12800*(x*e + d)*B*a^4*b^3*d^3*e^8 - 76800*(x*e + d)*A*a^3*
b^4*d^3*e^8 - 26880*B*a^4*b^3*d^4*e^8 - 26880*A*a^3*b^4*d^4*e^8 + 137995*(x*e + d)^3*B*a^5*b^2*e^9 - 413985*(x
*e + d)^3*A*a^4*b^3*e^9 - 49920*(x*e + d)^2*B*a^5*b^2*d*e^9 + 249600*(x*e + d)^2*A*a^4*b^3*d*e^9 - 3840*(x*e +
 d)*B*a^5*b^2*d^2*e^9 + 57600*(x*e + d)*A*a^4*b^3*d^2*e^9 + 16128*B*a^5*b^2*d^3*e^9 + 26880*A*a^4*b^3*d^3*e^9
+ 16640*(x*e + d)^2*B*a^6*b*e^10 - 49920*(x*e + d)^2*A*a^5*b^2*e^10 + 5120*(x*e + d)*B*a^6*b*d*e^10 - 23040*(x
*e + d)*A*a^5*b^2*d*e^10 - 5376*B*a^6*b*d^2*e^10 - 16128*A*a^5*b^2*d^2*e^10 - 1280*(x*e + d)*B*a^7*e^11 + 3840
*(x*e + d)*A*a^6*b*e^11 + 768*B*a^7*d*e^11 + 5376*A*a^6*b*d*e^11 - 768*A*a^7*e^12)/((b^8*d^8 - 8*a*b^7*d^7*e +
 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5 + 28*a^6*b^2*d^2*e^6 - 8*a^
7*b*d*e^7 + a^8*e^8)*((x*e + d)^(3/2)*b - sqrt(x*e + d)*b*d + sqrt(x*e + d)*a*e)^5)

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maple [B]  time = 0.10, size = 1735, normalized size = 3.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-20195/192*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d+5327/32*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(
e*x+d)^(1/2)*A*a*d^3-9443/128*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4+4253/192*e^5/(a*e-b*d)^8
*b^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d-749/5*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a*d^2-3269/64*e
^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d+36463/192*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/
2)*B*a*d^3+5327/32*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d-15981/64*e^7/(a*e-b*d)^8*b^5/(b*e*x
+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2+1029/16*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^2*d^3+1211/64*e^7/
(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^2+28329/64*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)
*A*a^2*d-4067/64*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^2*d^2+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a
*e)^5*(e*x+d)^(5/2)*A*a*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^2-252/5*e^6/(a*e-b*d)
^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d-2/3*e^5/(a*e-b*d)^7/(e*x+d)^(3/2)*a*B-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(
5/2)*A+4*e^5/(a*e-b*d)^7/(e*x+d)^(3/2)*A*b-42*e^5*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*A+2/5*e^4/(a*e-b*d)^6/(e*x+d)^
(5/2)*B*d+7837/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d+3003/64*e^4/(a*e-b*d)^8*b^3/((a*e-b*d)*b
)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d+1083/64*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2
)*B*d+350/3*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3-8099/96*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e
*x+d)^(3/2)*B*d^4-5327/128*e^9/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4-5327/128*e^5/(a*e-b*d)^8*b^7/
(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^4-9009/128*e^5/(a*e-b*d)^8*b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-
b*d)*b)^(1/2)*b)*A-3633/128*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A-10/3*e^4/(a*e-b*d)^7/(e*x+d)^(3/
2)*B*b*d+30*e^4*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*B*d+12*e^5*b/(a*e-b*d)^8/(e*x+d)^(1/2)*a*B+3003/128*e^5/(a*e-b*d
)^8*b^2/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B+1253/15*e^7/(a*e-b*d)^8*b^4/(b*e*x
+a*e)^5*(e*x+d)^(5/2)*B*a^3-9443/64*e^8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^3+9443/64*e^5/(a*e-b*d
)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*d^3+12131/192*e^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4+9629
/192*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2+1467/128*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^
(9/2)*B*a-1001/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^2-1001/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^
5*(e*x+d)^(5/2)*A*d^2+2373/128*e^9/(a*e-b*d)^8*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+1477/64*e^4/(a*e-b*d)^8*b
^7/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*d^5-6941/96*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2-7837/64*e^6
/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*a

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(7/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*e-b*d>0)', see `assume?` for
 more details)Is a*e-b*d positive or negative?

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mupad [B]  time = 3.50, size = 802, normalized size = 1.84 \[ \frac {\frac {27599\,{\left (d+e\,x\right )}^3\,\left (-3\,A\,b^3\,e^5+2\,B\,d\,b^3\,e^4+B\,a\,b^2\,e^5\right )}{384\,{\left (a\,e-b\,d\right )}^4}-\frac {2\,\left (A\,e^5-B\,d\,e^4\right )}{5\,\left (a\,e-b\,d\right )}+\frac {11297\,{\left (d+e\,x\right )}^4\,\left (-3\,A\,b^4\,e^5+2\,B\,d\,b^4\,e^4+B\,a\,b^3\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {7007\,{\left (d+e\,x\right )}^6\,\left (-3\,A\,b^6\,e^5+2\,B\,d\,b^6\,e^4+B\,a\,b^5\,e^5\right )}{64\,{\left (a\,e-b\,d\right )}^7}-\frac {2\,\left (d+e\,x\right )\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^2}+\frac {26\,b\,{\left (d+e\,x\right )}^2\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{3\,{\left (a\,e-b\,d\right )}^3}+\frac {1001\,b^4\,{\left (d+e\,x\right )}^5\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{5\,{\left (a\,e-b\,d\right )}^6}+\frac {3003\,b^6\,{\left (d+e\,x\right )}^7\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^8}}{{\left (d+e\,x\right )}^{5/2}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{9/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{7/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{15/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{13/2}+{\left (d+e\,x\right )}^{11/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}+\frac {3003\,b^{3/2}\,e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )\,\left (a^8\,e^8-8\,a^7\,b\,d\,e^7+28\,a^6\,b^2\,d^2\,e^6-56\,a^5\,b^3\,d^3\,e^5+70\,a^4\,b^4\,d^4\,e^4-56\,a^3\,b^5\,d^5\,e^3+28\,a^2\,b^6\,d^6\,e^2-8\,a\,b^7\,d^7\,e+b^8\,d^8\right )}{{\left (a\,e-b\,d\right )}^{17/2}\,\left (B\,a\,e^5-3\,A\,b\,e^5+2\,B\,b\,d\,e^4\right )}\right )\,\left (B\,a\,e-3\,A\,b\,e+2\,B\,b\,d\right )}{128\,{\left (a\,e-b\,d\right )}^{17/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)/((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3),x)

[Out]

((27599*(d + e*x)^3*(B*a*b^2*e^5 - 3*A*b^3*e^5 + 2*B*b^3*d*e^4))/(384*(a*e - b*d)^4) - (2*(A*e^5 - B*d*e^4))/(
5*(a*e - b*d)) + (11297*(d + e*x)^4*(B*a*b^3*e^5 - 3*A*b^4*e^5 + 2*B*b^4*d*e^4))/(64*(a*e - b*d)^5) + (7007*(d
 + e*x)^6*(B*a*b^5*e^5 - 3*A*b^6*e^5 + 2*B*b^6*d*e^4))/(64*(a*e - b*d)^7) - (2*(d + e*x)*(B*a*e^5 - 3*A*b*e^5
+ 2*B*b*d*e^4))/(3*(a*e - b*d)^2) + (26*b*(d + e*x)^2*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(3*(a*e - b*d)^3) +
 (1001*b^4*(d + e*x)^5*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4))/(5*(a*e - b*d)^6) + (3003*b^6*(d + e*x)^7*(B*a*e^5
 - 3*A*b*e^5 + 2*B*b*d*e^4))/(128*(a*e - b*d)^8))/((d + e*x)^(5/2)*(a^5*e^5 - b^5*d^5 - 10*a^2*b^3*d^3*e^2 + 1
0*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4) - (d + e*x)^(9/2)*(10*b^5*d^3 - 10*a^3*b^2*e^3 + 30*a^2*b^3
*d*e^2 - 30*a*b^4*d^2*e) + (d + e*x)^(7/2)*(5*b^5*d^4 + 5*a^4*b*e^4 - 20*a^3*b^2*d*e^3 + 30*a^2*b^3*d^2*e^2 -
20*a*b^4*d^3*e) + b^5*(d + e*x)^(15/2) - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^(13/2) + (d + e*x)^(11/2)*(10*b^5*d^2
 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e)) + (3003*b^(3/2)*e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(B*a*e - 3*A*b*e + 2*
B*b*d)*(a^8*e^8 + b^8*d^8 + 28*a^2*b^6*d^6*e^2 - 56*a^3*b^5*d^5*e^3 + 70*a^4*b^4*d^4*e^4 - 56*a^5*b^3*d^3*e^5
+ 28*a^6*b^2*d^2*e^6 - 8*a*b^7*d^7*e - 8*a^7*b*d*e^7))/((a*e - b*d)^(17/2)*(B*a*e^5 - 3*A*b*e^5 + 2*B*b*d*e^4)
))*(B*a*e - 3*A*b*e + 2*B*b*d))/(128*(a*e - b*d)^(17/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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