3.1817 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a^2+2 a b x+b^2 x^2)^2} \, dx\)

Optimal. Leaf size=284 \[ -\frac {35 e^2 \sqrt {b d-a e} (-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}+\frac {35 e^2 \sqrt {d+e x} (-3 a B e+A b e+2 b B d)}{8 b^5}+\frac {35 e^2 (d+e x)^{3/2} (-3 a B e+A b e+2 b B d)}{24 b^4 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-3 a B e+A b e+2 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{7/2} (-3 a B e+A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

[Out]

35/24*e^2*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(3/2)/b^4/(-a*e+b*d)-7/8*e*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(5/2)/b^3
/(-a*e+b*d)/(b*x+a)-1/4*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(7/2)/b^2/(-a*e+b*d)/(b*x+a)^2-1/3*(A*b-B*a)*(e*x+d)^(
9/2)/b/(-a*e+b*d)/(b*x+a)^3-35/8*e^2*(A*b*e-3*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)^(1/2)/b^(11/2)+35/8*e^2*(A*b*e-3*B*a*e+2*B*b*d)*(e*x+d)^(1/2)/b^5

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Rubi [A]  time = 0.24, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {27, 78, 47, 50, 63, 208} \[ \frac {35 e^2 (d+e x)^{3/2} (-3 a B e+A b e+2 b B d)}{24 b^4 (b d-a e)}+\frac {35 e^2 \sqrt {d+e x} (-3 a B e+A b e+2 b B d)}{8 b^5}-\frac {35 e^2 \sqrt {b d-a e} (-3 a B e+A b e+2 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}-\frac {(d+e x)^{7/2} (-3 a B e+A b e+2 b B d)}{4 b^2 (a+b x)^2 (b d-a e)}-\frac {7 e (d+e x)^{5/2} (-3 a B e+A b e+2 b B d)}{8 b^3 (a+b x) (b d-a e)}-\frac {(d+e x)^{9/2} (A b-a B)}{3 b (a+b x)^3 (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*e^2*(2*b*B*d + A*b*e - 3*a*B*e)*Sqrt[d + e*x])/(8*b^5) + (35*e^2*(2*b*B*d + A*b*e - 3*a*B*e)*(d + e*x)^(3/
2))/(24*b^4*(b*d - a*e)) - (7*e*(2*b*B*d + A*b*e - 3*a*B*e)*(d + e*x)^(5/2))/(8*b^3*(b*d - a*e)*(a + b*x)) - (
(2*b*B*d + A*b*e - 3*a*B*e)*(d + e*x)^(7/2))/(4*b^2*(b*d - a*e)*(a + b*x)^2) - ((A*b - a*B)*(d + e*x)^(9/2))/(
3*b*(b*d - a*e)*(a + b*x)^3) - (35*e^2*Sqrt[b*d - a*e]*(2*b*B*d + A*b*e - 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e
*x])/Sqrt[b*d - a*e]])/(8*b^(11/2))

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 47

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 )^2} \, dx &=\int \frac {(A+B x) (d+e x)^{7/2}}{(a+b x)^4} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(2 b B d+A b e-3 a B e) \int \frac {(d+e x)^{7/2}}{(a+b x)^3} \, dx}{2 b (b d-a e)}\\ &=-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(7 e (2 b B d+A b e-3 a B e)) \int \frac {(d+e x)^{5/2}}{(a+b x)^2} \, dx}{8 b^2 (b d-a e)}\\ &=-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (35 e^2 (2 b B d+A b e-3 a B e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{16 b^3 (b d-a e)}\\ &=\frac {35 e^2 (2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{24 b^4 (b d-a e)}-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (35 e^2 (2 b B d+A b e-3 a B e)\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^4}\\ &=\frac {35 e^2 (2 b B d+A b e-3 a B e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^2 (2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{24 b^4 (b d-a e)}-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {\left (35 e^2 (b d-a e) (2 b B d+A b e-3 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^5}\\ &=\frac {35 e^2 (2 b B d+A b e-3 a B e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^2 (2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{24 b^4 (b d-a e)}-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}+\frac {(35 e (b d-a e) (2 b B d+A b e-3 a B e)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^5}\\ &=\frac {35 e^2 (2 b B d+A b e-3 a B e) \sqrt {d+e x}}{8 b^5}+\frac {35 e^2 (2 b B d+A b e-3 a B e) (d+e x)^{3/2}}{24 b^4 (b d-a e)}-\frac {7 e (2 b B d+A b e-3 a B e) (d+e x)^{5/2}}{8 b^3 (b d-a e) (a+b x)}-\frac {(2 b B d+A b e-3 a B e) (d+e x)^{7/2}}{4 b^2 (b d-a e) (a+b x)^2}-\frac {(A b-a B) (d+e x)^{9/2}}{3 b (b d-a e) (a+b x)^3}-\frac {35 e^2 \sqrt {b d-a e} (2 b B d+A b e-3 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{11/2}}\\ \end {align*}

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Mathematica [C]  time = 0.09, size = 99, normalized size = 0.35 \[ \frac {(d+e x)^{9/2} \left (\frac {9 a B-9 A b}{(a+b x)^3}-\frac {3 e^2 (-3 a B e+A b e+2 b B d) \, _2F_1\left (3,\frac {9}{2};\frac {11}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^3}\right )}{27 b (b d-a e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(9/2)*((-9*A*b + 9*a*B)/(a + b*x)^3 - (3*e^2*(2*b*B*d + A*b*e - 3*a*B*e)*Hypergeometric2F1[3, 9/2,
11/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^3))/(27*b*(b*d - a*e))

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fricas [B]  time = 1.18, size = 1026, normalized size = 3.61 \[ \left [-\frac {105 \, {\left (2 \, B a^{3} b d e^{2} - {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d e^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (2 \, B a b^{3} d e^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{2} b^{2} d e^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (16 \, B b^{4} e^{3} x^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 14 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 35 \, {\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + 16 \, {\left (10 \, B b^{4} d e^{2} - 3 \, {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} - 3 \, {\left (26 \, B b^{4} d^{2} e - {\left (241 \, B a b^{3} - 29 \, A b^{4}\right )} d e^{2} + 77 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (6 \, B b^{4} d^{3} + {\left (41 \, B a b^{3} + 19 \, A b^{4}\right )} d^{2} e - 7 \, {\left (61 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} + 140 \, {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}, -\frac {105 \, {\left (2 \, B a^{3} b d e^{2} - {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + {\left (2 \, B b^{4} d e^{2} - {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} + 3 \, {\left (2 \, B a b^{3} d e^{2} - {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} + 3 \, {\left (2 \, B a^{2} b^{2} d e^{2} - {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (16 \, B b^{4} e^{3} x^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} - 14 \, {\left (2 \, B a^{2} b^{2} + A a b^{3}\right )} d^{2} e + 35 \, {\left (9 \, B a^{3} b - A a^{2} b^{2}\right )} d e^{2} - 105 \, {\left (3 \, B a^{4} - A a^{3} b\right )} e^{3} + 16 \, {\left (10 \, B b^{4} d e^{2} - 3 \, {\left (3 \, B a b^{3} - A b^{4}\right )} e^{3}\right )} x^{3} - 3 \, {\left (26 \, B b^{4} d^{2} e - {\left (241 \, B a b^{3} - 29 \, A b^{4}\right )} d e^{2} + 77 \, {\left (3 \, B a^{2} b^{2} - A a b^{3}\right )} e^{3}\right )} x^{2} - 2 \, {\left (6 \, B b^{4} d^{3} + {\left (41 \, B a b^{3} + 19 \, A b^{4}\right )} d^{2} e - 7 \, {\left (61 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} d e^{2} + 140 \, {\left (3 \, B a^{3} b - A a^{2} b^{2}\right )} e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[-1/48*(105*(2*B*a^3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a*b^3 - A*b^4)*e^3)*x^3 + 3*(2*
B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(2*B*a^2*b^2*d*e^2 - (3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt
((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e + 2*sqrt(e*x + d)*b*sqrt((b*d - a*e)/b))/(b*x + a)) - 2*(16*B*b^4*e^3
*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3 - 14*(2*B*a^2*b^2 + A*a*b^3)*d^2*e + 35*(9*B*a^3*b - A*a^2*b^2)*d*e^2 - 105*(
3*B*a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3*(26*B*b^4*d^2*e - (241*B*a*b^
3 - 29*A*b^4)*d*e^2 + 77*(3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 - 2*(6*B*b^4*d^3 + (41*B*a*b^3 + 19*A*b^4)*d^2*e - 7
*(61*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 + 140*(3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7*x^2
+ 3*a^2*b^6*x + a^3*b^5), -1/24*(105*(2*B*a^3*b*d*e^2 - (3*B*a^4 - A*a^3*b)*e^3 + (2*B*b^4*d*e^2 - (3*B*a*b^3
- A*b^4)*e^3)*x^3 + 3*(2*B*a*b^3*d*e^2 - (3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 + 3*(2*B*a^2*b^2*d*e^2 - (3*B*a^3*b
- A*a^2*b^2)*e^3)*x)*sqrt(-(b*d - a*e)/b)*arctan(-sqrt(e*x + d)*b*sqrt(-(b*d - a*e)/b)/(b*d - a*e)) - (16*B*b^
4*e^3*x^4 - 4*(B*a*b^3 + 2*A*b^4)*d^3 - 14*(2*B*a^2*b^2 + A*a*b^3)*d^2*e + 35*(9*B*a^3*b - A*a^2*b^2)*d*e^2 -
105*(3*B*a^4 - A*a^3*b)*e^3 + 16*(10*B*b^4*d*e^2 - 3*(3*B*a*b^3 - A*b^4)*e^3)*x^3 - 3*(26*B*b^4*d^2*e - (241*B
*a*b^3 - 29*A*b^4)*d*e^2 + 77*(3*B*a^2*b^2 - A*a*b^3)*e^3)*x^2 - 2*(6*B*b^4*d^3 + (41*B*a*b^3 + 19*A*b^4)*d^2*
e - 7*(61*B*a^2*b^2 - 7*A*a*b^3)*d*e^2 + 140*(3*B*a^3*b - A*a^2*b^2)*e^3)*x)*sqrt(e*x + d))/(b^8*x^3 + 3*a*b^7
*x^2 + 3*a^2*b^6*x + a^3*b^5)]

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giac [B]  time = 0.25, size = 577, normalized size = 2.03 \[ \frac {35 \, {\left (2 \, B b^{2} d^{2} e^{2} - 5 \, B a b d e^{3} + A b^{2} d e^{3} + 3 \, B a^{2} e^{4} - A a b e^{4}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {78 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{4} d^{2} e^{2} - 144 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{4} d^{3} e^{2} + 66 \, \sqrt {x e + d} B b^{4} d^{4} e^{2} - 243 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{3} d e^{3} + 87 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{4} d e^{3} + 568 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{3} d^{2} e^{3} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{4} d^{2} e^{3} - 321 \, \sqrt {x e + d} B a b^{3} d^{3} e^{3} + 57 \, \sqrt {x e + d} A b^{4} d^{3} e^{3} + 165 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{2} e^{4} - 87 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{3} e^{4} - 704 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{2} d e^{4} + 272 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{3} d e^{4} + 567 \, \sqrt {x e + d} B a^{2} b^{2} d^{2} e^{4} - 171 \, \sqrt {x e + d} A a b^{3} d^{2} e^{4} + 280 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b e^{5} - 136 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{2} e^{5} - 435 \, \sqrt {x e + d} B a^{3} b d e^{5} + 171 \, \sqrt {x e + d} A a^{2} b^{2} d e^{5} + 123 \, \sqrt {x e + d} B a^{4} e^{6} - 57 \, \sqrt {x e + d} A a^{3} b e^{6}}{24 \, {\left ({\left (x e + d\right )} b - b d + a e\right )}^{3} b^{5}} + \frac {2 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{8} e^{2} + 9 \, \sqrt {x e + d} B b^{8} d e^{2} - 12 \, \sqrt {x e + d} B a b^{7} e^{3} + 3 \, \sqrt {x e + d} A b^{8} e^{3}\right )}}{3 \, b^{12}} \]

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)^2,x, algorithm="giac")

[Out]

35/8*(2*B*b^2*d^2*e^2 - 5*B*a*b*d*e^3 + A*b^2*d*e^3 + 3*B*a^2*e^4 - A*a*b*e^4)*arctan(sqrt(x*e + d)*b/sqrt(-b^
2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^5) - 1/24*(78*(x*e + d)^(5/2)*B*b^4*d^2*e^2 - 144*(x*e + d)^(3/2)*B*b^4*
d^3*e^2 + 66*sqrt(x*e + d)*B*b^4*d^4*e^2 - 243*(x*e + d)^(5/2)*B*a*b^3*d*e^3 + 87*(x*e + d)^(5/2)*A*b^4*d*e^3
+ 568*(x*e + d)^(3/2)*B*a*b^3*d^2*e^3 - 136*(x*e + d)^(3/2)*A*b^4*d^2*e^3 - 321*sqrt(x*e + d)*B*a*b^3*d^3*e^3
+ 57*sqrt(x*e + d)*A*b^4*d^3*e^3 + 165*(x*e + d)^(5/2)*B*a^2*b^2*e^4 - 87*(x*e + d)^(5/2)*A*a*b^3*e^4 - 704*(x
*e + d)^(3/2)*B*a^2*b^2*d*e^4 + 272*(x*e + d)^(3/2)*A*a*b^3*d*e^4 + 567*sqrt(x*e + d)*B*a^2*b^2*d^2*e^4 - 171*
sqrt(x*e + d)*A*a*b^3*d^2*e^4 + 280*(x*e + d)^(3/2)*B*a^3*b*e^5 - 136*(x*e + d)^(3/2)*A*a^2*b^2*e^5 - 435*sqrt
(x*e + d)*B*a^3*b*d*e^5 + 171*sqrt(x*e + d)*A*a^2*b^2*d*e^5 + 123*sqrt(x*e + d)*B*a^4*e^6 - 57*sqrt(x*e + d)*A
*a^3*b*e^6)/(((x*e + d)*b - b*d + a*e)^3*b^5) + 2/3*((x*e + d)^(3/2)*B*b^8*e^2 + 9*sqrt(x*e + d)*B*b^8*d*e^2 -
12*sqrt(x*e + d)*B*a*b^7*e^3 + 3*sqrt(x*e + d)*A*b^8*e^3)/b^12

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maple [B]  time = 0.08, size = 905, normalized size = 3.19 \[ \frac {19 \sqrt {e x +d}\, A \,a^{3} e^{6}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {57 \sqrt {e x +d}\, A \,a^{2} d \,e^{5}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {57 \sqrt {e x +d}\, A a \,d^{2} e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {19 \sqrt {e x +d}\, A \,d^{3} e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {41 \sqrt {e x +d}\, B \,a^{4} e^{6}}{8 \left (b e x +a e \right )^{3} b^{5}}+\frac {145 \sqrt {e x +d}\, B \,a^{3} d \,e^{5}}{8 \left (b e x +a e \right )^{3} b^{4}}-\frac {189 \sqrt {e x +d}\, B \,a^{2} d^{2} e^{4}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {107 \sqrt {e x +d}\, B a \,d^{3} e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {11 \sqrt {e x +d}\, B \,d^{4} e^{2}}{4 \left (b e x +a e \right )^{3} b}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} A \,a^{2} e^{5}}{3 \left (b e x +a e \right )^{3} b^{3}}-\frac {34 \left (e x +d \right )^{\frac {3}{2}} A a d \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {17 \left (e x +d \right )^{\frac {3}{2}} A \,d^{2} e^{3}}{3 \left (b e x +a e \right )^{3} b}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} B \,a^{3} e^{5}}{3 \left (b e x +a e \right )^{3} b^{4}}+\frac {88 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} d \,e^{4}}{3 \left (b e x +a e \right )^{3} b^{3}}-\frac {71 \left (e x +d \right )^{\frac {3}{2}} B a \,d^{2} e^{3}}{3 \left (b e x +a e \right )^{3} b^{2}}+\frac {6 \left (e x +d \right )^{\frac {3}{2}} B \,d^{3} e^{2}}{\left (b e x +a e \right )^{3} b}+\frac {29 \left (e x +d \right )^{\frac {5}{2}} A a \,e^{4}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {29 \left (e x +d \right )^{\frac {5}{2}} A d \,e^{3}}{8 \left (b e x +a e \right )^{3} b}-\frac {55 \left (e x +d \right )^{\frac {5}{2}} B \,a^{2} e^{4}}{8 \left (b e x +a e \right )^{3} b^{3}}+\frac {81 \left (e x +d \right )^{\frac {5}{2}} B a d \,e^{3}}{8 \left (b e x +a e \right )^{3} b^{2}}-\frac {13 \left (e x +d \right )^{\frac {5}{2}} B \,d^{2} e^{2}}{4 \left (b e x +a e \right )^{3} b}-\frac {35 A a \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {35 A d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {105 B \,a^{2} e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{5}}-\frac {175 B a d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4}}+\frac {35 B \,d^{2} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{4 \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {2 \sqrt {e x +d}\, A \,e^{3}}{b^{4}}-\frac {8 \sqrt {e x +d}\, B a \,e^{3}}{b^{5}}+\frac {6 \sqrt {e x +d}\, B d \,e^{2}}{b^{4}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} B \,e^{2}}{3 b^{4}} \]

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)^2,x)

[Out]

2/3*e^2/b^4*B*(e*x+d)^(3/2)+2*e^3/b^4*A*(e*x+d)^(1/2)-8*e^3/b^5*a*B*(e*x+d)^(1/2)+6*e^2/b^4*B*d*(e*x+d)^(1/2)-
34/3*e^4/b^2/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a*d+88/3*e^4/b^3/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^2*d-71/3*e^3/b^2/(
b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a*d^2+145/8*e^5/b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d*a^3-57/8*e^5/b^3/(b*e*x+a*e)^
3*(e*x+d)^(1/2)*A*a^2*d+57/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2*a+81/8*e^3/b^2/(b*e*x+a*e)^3*(e*x+d)^(5
/2)*B*a*d-175/8*e^3/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a*d+107/8*e^3/b^2/(b
*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^3-189/8*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d^2-41/8*e^6/b^5/(b*e*x+a*e)
^3*(e*x+d)^(1/2)*B*a^4+35/8*e^3/b^3/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A*d+105/8*
e^4/b^5/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*a^2-13/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)
^(5/2)*B*d^2+6*e^2/b/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^3-11/4*e^2/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^4+35/4*e^2/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^2+29/8*e^4/b^2/(b*e*x+a*e)^3*(e*x+d)^(5
/2)*A*a-29/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A*d-55/8*e^4/b^3/(b*e*x+a*e)^3*(e*x+d)^(5/2)*B*a^2+17/3*e^5/b^3
/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a^2-35/8*e^4/b^4/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b
)*A*a+17/3*e^3/b/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d^2-35/3*e^5/b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*a^3+19/8*e^6/b^4
/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^3-19/8*e^3/b/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^3

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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)^2,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 = 2.16, size = 513, normalized size = 1.81 \[ \left (\frac {2\,A\,e^3-2\,B\,d\,e^2}{b^4}+\frac {2\,B\,e^2\,\left (4\,b^4\,d-4\,a\,b^3\,e\right )}{b^8}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (\frac {41\,B\,a^4\,e^6}{8}-\frac {145\,B\,a^3\,b\,d\,e^5}{8}-\frac {19\,A\,a^3\,b\,e^6}{8}+\frac {189\,B\,a^2\,b^2\,d^2\,e^4}{8}+\frac {57\,A\,a^2\,b^2\,d\,e^5}{8}-\frac {107\,B\,a\,b^3\,d^3\,e^3}{8}-\frac {57\,A\,a\,b^3\,d^2\,e^4}{8}+\frac {11\,B\,b^4\,d^4\,e^2}{4}+\frac {19\,A\,b^4\,d^3\,e^3}{8}\right )-{\left (d+e\,x\right )}^{3/2}\,\left (-\frac {35\,B\,a^3\,b\,e^5}{3}+\frac {88\,B\,a^2\,b^2\,d\,e^4}{3}+\frac {17\,A\,a^2\,b^2\,e^5}{3}-\frac {71\,B\,a\,b^3\,d^2\,e^3}{3}-\frac {34\,A\,a\,b^3\,d\,e^4}{3}+6\,B\,b^4\,d^3\,e^2+\frac {17\,A\,b^4\,d^2\,e^3}{3}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {55\,B\,a^2\,b^2\,e^4}{8}-\frac {81\,B\,a\,b^3\,d\,e^3}{8}-\frac {29\,A\,a\,b^3\,e^4}{8}+\frac {13\,B\,b^4\,d^2\,e^2}{4}+\frac {29\,A\,b^4\,d\,e^3}{8}\right )}{b^8\,{\left (d+e\,x\right )}^3-\left (3\,b^8\,d-3\,a\,b^7\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^6\,e^2-6\,a\,b^7\,d\,e+3\,b^8\,d^2\right )-b^8\,d^3+a^3\,b^5\,e^3-3\,a^2\,b^6\,d\,e^2+3\,a\,b^7\,d^2\,e}+\frac {2\,B\,e^2\,{\left (d+e\,x\right )}^{3/2}}{3\,b^4}+\frac {e^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,1{}\mathrm {i}}{\sqrt {b\,d-a\,e}}\right )\,\sqrt {b\,d-a\,e}\,\left (A\,b\,e-3\,B\,a\,e+2\,B\,b\,d\right )\,35{}\mathrm {i}}{8\,b^{11/2}} \]

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)^2,x)

[Out]

((2*A*e^3 - 2*B*d*e^2)/b^4 + (2*B*e^2*(4*b^4*d - 4*a*b^3*e))/b^8)*(d + e*x)^(1/2) - ((d + e*x)^(1/2)*((41*B*a^
4*e^6)/8 - (19*A*a^3*b*e^6)/8 + (19*A*b^4*d^3*e^3)/8 + (11*B*b^4*d^4*e^2)/4 - (57*A*a*b^3*d^2*e^4)/8 + (57*A*a
^2*b^2*d*e^5)/8 - (107*B*a*b^3*d^3*e^3)/8 + (189*B*a^2*b^2*d^2*e^4)/8 - (145*B*a^3*b*d*e^5)/8) - (d + e*x)^(3/
2)*((17*A*a^2*b^2*e^5)/3 - (35*B*a^3*b*e^5)/3 + (17*A*b^4*d^2*e^3)/3 + 6*B*b^4*d^3*e^2 - (71*B*a*b^3*d^2*e^3)/
3 + (88*B*a^2*b^2*d*e^4)/3 - (34*A*a*b^3*d*e^4)/3) + (d + e*x)^(5/2)*((29*A*b^4*d*e^3)/8 - (29*A*a*b^3*e^4)/8
+ (55*B*a^2*b^2*e^4)/8 + (13*B*b^4*d^2*e^2)/4 - (81*B*a*b^3*d*e^3)/8))/(b^8*(d + e*x)^3 - (3*b^8*d - 3*a*b^7*e
)*(d + e*x)^2 + (d + e*x)*(3*b^8*d^2 + 3*a^2*b^6*e^2 - 6*a*b^7*d*e) - b^8*d^3 + a^3*b^5*e^3 - 3*a^2*b^6*d*e^2
+ 3*a*b^7*d^2*e) + (2*B*e^2*(d + e*x)^(3/2))/(3*b^4) + (e^2*atan((b^(1/2)*(d + e*x)^(1/2)*1i)/(b*d - a*e)^(1/2
))*(b*d - a*e)^(1/2)*(A*b*e - 3*B*a*e + 2*B*b*d)*35i)/(8*b^(11/2))

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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)**2,x)

[Out]

Timed out

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