3.17.8 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a^2+2 a b x+b^2 x^2)^3} \, dx\)

Optimal. Leaf size=313 \[ \frac {e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac {e^3 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac {e^2 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}-\frac {e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \]

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Rubi [A]  time = 0.27, antiderivative size = 313, 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, 51, 63, 208} \begin {gather*} -\frac {e^3 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{128 b^4 (a+b x) (b d-a e)^2}-\frac {e^2 \sqrt {d+e x} (-7 a B e-3 A b e+10 b B d)}{64 b^4 (a+b x)^2 (b d-a e)}+\frac {e^4 (-7 a B e-3 A b e+10 b B d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}-\frac {e (d+e x)^{3/2} (-7 a B e-3 A b e+10 b B d)}{48 b^3 (a+b x)^3 (b d-a e)}-\frac {(d+e x)^{5/2} (-7 a B e-3 A b e+10 b B d)}{40 b^2 (a+b x)^4 (b d-a e)}-\frac {(d+e x)^{7/2} (A b-a B)}{5 b (a+b x)^5 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

-(e^2*(10*b*B*d - 3*A*b*e - 7*a*B*e)*Sqrt[d + e*x])/(64*b^4*(b*d - a*e)*(a + b*x)^2) - (e^3*(10*b*B*d - 3*A*b*
e - 7*a*B*e)*Sqrt[d + e*x])/(128*b^4*(b*d - a*e)^2*(a + b*x)) - (e*(10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(3
/2))/(48*b^3*(b*d - a*e)*(a + b*x)^3) - ((10*b*B*d - 3*A*b*e - 7*a*B*e)*(d + e*x)^(5/2))/(40*b^2*(b*d - a*e)*(
a + b*x)^4) - ((A*b - a*B)*(d + e*x)^(7/2))/(5*b*(b*d - a*e)*(a + b*x)^5) + (e^4*(10*b*B*d - 3*A*b*e - 7*a*B*e
)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*b^(9/2)*(b*d - a*e)^(5/2))

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 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)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx &=\int \frac {(A+B x) (d+e x)^{5/2}}{(a+b x)^6} \, dx\\ &=-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(10 b B d-3 A b e-7 a B e) \int \frac {(d+e x)^{5/2}}{(a+b x)^5} \, dx}{10 b (b d-a e)}\\ &=-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {(e (10 b B d-3 A b e-7 a B e)) \int \frac {(d+e x)^{3/2}}{(a+b x)^4} \, dx}{16 b^2 (b d-a e)}\\ &=-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^2 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^3} \, dx}{32 b^3 (b d-a e)}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {\left (e^3 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^4 (b d-a e)}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^4 (10 b B d-3 A b e-7 a B e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^4 (b d-a e)^2}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}-\frac {\left (e^3 (10 b B d-3 A b e-7 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^4 (b d-a e)^2}\\ &=-\frac {e^2 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{64 b^4 (b d-a e) (a+b x)^2}-\frac {e^3 (10 b B d-3 A b e-7 a B e) \sqrt {d+e x}}{128 b^4 (b d-a e)^2 (a+b x)}-\frac {e (10 b B d-3 A b e-7 a B e) (d+e x)^{3/2}}{48 b^3 (b d-a e) (a+b x)^3}-\frac {(10 b B d-3 A b e-7 a B e) (d+e x)^{5/2}}{40 b^2 (b d-a e) (a+b x)^4}-\frac {(A b-a B) (d+e x)^{7/2}}{5 b (b d-a e) (a+b x)^5}+\frac {e^4 (10 b B d-3 A b e-7 a B e) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{9/2} (b d-a e)^{5/2}}\\ \end {align*}

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Mathematica [C]  time = 0.06, size = 99, normalized size = 0.32 \begin {gather*} \frac {(d+e x)^{7/2} \left (\frac {e^4 (7 a B e+3 A b e-10 b B d) \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {b (d+e x)}{b d-a e}\right )}{(b d-a e)^5}+\frac {7 a B-7 A b}{(a+b x)^5}\right )}{35 b (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

((d + e*x)^(7/2)*((-7*A*b + 7*a*B)/(a + b*x)^5 + (e^4*(-10*b*B*d + 3*A*b*e + 7*a*B*e)*Hypergeometric2F1[7/2, 5
, 9/2, (b*(d + e*x))/(b*d - a*e)])/(b*d - a*e)^5))/(35*b*(b*d - a*e))

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IntegrateAlgebraic [B]  time = 4.64, size = 676, normalized size = 2.16 \begin {gather*} \frac {e^4 \sqrt {d+e x} \left (105 a^5 B e^5+45 a^4 A b e^5+490 a^4 b B e^4 (d+e x)-570 a^4 b B d e^4+210 a^3 A b^2 e^4 (d+e x)-180 a^3 A b^2 d e^4+1230 a^3 b^2 B d^2 e^3+896 a^3 b^2 B e^3 (d+e x)^2-2170 a^3 b^2 B d e^3 (d+e x)+270 a^2 A b^3 d^2 e^3+384 a^2 A b^3 e^3 (d+e x)^2-630 a^2 A b^3 d e^3 (d+e x)-1320 a^2 b^3 B d^3 e^2+3570 a^2 b^3 B d^2 e^2 (d+e x)+790 a^2 b^3 B e^2 (d+e x)^3-3072 a^2 b^3 B d e^2 (d+e x)^2-180 a A b^4 d^3 e^2+630 a A b^4 d^2 e^2 (d+e x)-210 a A b^4 e^2 (d+e x)^3-768 a A b^4 d e^2 (d+e x)^2+705 a b^4 B d^4 e-2590 a b^4 B d^3 e (d+e x)+3456 a b^4 B d^2 e (d+e x)^2-105 a b^4 B e (d+e x)^4-1370 a b^4 B d e (d+e x)^3+45 A b^5 d^4 e-210 A b^5 d^3 e (d+e x)+384 A b^5 d^2 e (d+e x)^2-45 A b^5 e (d+e x)^4+210 A b^5 d e (d+e x)^3-150 b^5 B d^5+700 b^5 B d^4 (d+e x)-1280 b^5 B d^3 (d+e x)^2+580 b^5 B d^2 (d+e x)^3+150 b^5 B d (d+e x)^4\right )}{1920 b^4 (b d-a e)^2 (-a e-b (d+e x)+b d)^5}+\frac {\left (-7 a B e^5-3 A b e^5+10 b B d e^4\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x} \sqrt {a e-b d}}{b d-a e}\right )}{128 b^{9/2} (b d-a e)^2 \sqrt {a e-b d}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(a^2 + 2*a*b*x + b^2*x^2)^3,x]

[Out]

(e^4*Sqrt[d + e*x]*(-150*b^5*B*d^5 + 45*A*b^5*d^4*e + 705*a*b^4*B*d^4*e - 180*a*A*b^4*d^3*e^2 - 1320*a^2*b^3*B
*d^3*e^2 + 270*a^2*A*b^3*d^2*e^3 + 1230*a^3*b^2*B*d^2*e^3 - 180*a^3*A*b^2*d*e^4 - 570*a^4*b*B*d*e^4 + 45*a^4*A
*b*e^5 + 105*a^5*B*e^5 + 700*b^5*B*d^4*(d + e*x) - 210*A*b^5*d^3*e*(d + e*x) - 2590*a*b^4*B*d^3*e*(d + e*x) +
630*a*A*b^4*d^2*e^2*(d + e*x) + 3570*a^2*b^3*B*d^2*e^2*(d + e*x) - 630*a^2*A*b^3*d*e^3*(d + e*x) - 2170*a^3*b^
2*B*d*e^3*(d + e*x) + 210*a^3*A*b^2*e^4*(d + e*x) + 490*a^4*b*B*e^4*(d + e*x) - 1280*b^5*B*d^3*(d + e*x)^2 + 3
84*A*b^5*d^2*e*(d + e*x)^2 + 3456*a*b^4*B*d^2*e*(d + e*x)^2 - 768*a*A*b^4*d*e^2*(d + e*x)^2 - 3072*a^2*b^3*B*d
*e^2*(d + e*x)^2 + 384*a^2*A*b^3*e^3*(d + e*x)^2 + 896*a^3*b^2*B*e^3*(d + e*x)^2 + 580*b^5*B*d^2*(d + e*x)^3 +
210*A*b^5*d*e*(d + e*x)^3 - 1370*a*b^4*B*d*e*(d + e*x)^3 - 210*a*A*b^4*e^2*(d + e*x)^3 + 790*a^2*b^3*B*e^2*(d
+ e*x)^3 + 150*b^5*B*d*(d + e*x)^4 - 45*A*b^5*e*(d + e*x)^4 - 105*a*b^4*B*e*(d + e*x)^4))/(1920*b^4*(b*d - a*
e)^2*(b*d - a*e - b*(d + e*x))^5) + ((10*b*B*d*e^4 - 3*A*b*e^5 - 7*a*B*e^5)*ArcTan[(Sqrt[b]*Sqrt[-(b*d) + a*e]
*Sqrt[d + e*x])/(b*d - a*e)])/(128*b^(9/2)*(b*d - a*e)^2*Sqrt[-(b*d) + a*e])

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fricas [B]  time = 0.50, size = 2238, normalized size = 7.15

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")

[Out]

[-1/3840*(15*(10*B*a^5*b*d*e^4 - (7*B*a^6 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5
+ 5*(10*B*a*b^5*d*e^4 - (7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*b^3 + 3*A*a^2*b
^4)*e^5)*x^3 + 10*(10*B*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^
5*b + 3*A*a^4*b^2)*e^5)*x)*sqrt(b^2*d - a*b*e)*log((b*e*x + 2*b*d - a*e - 2*sqrt(b^2*d - a*b*e)*sqrt(e*x + d))
/(b*x + a)) + 2*(96*(B*a*b^6 + 4*A*b^7)*d^5 - 16*(8*B*a^2*b^5 + 57*A*a*b^6)*d^4*e - 12*(B*a^3*b^4 - 46*A*a^2*b
^5)*d^3*e^2 - 6*(6*B*a^4*b^3 - A*a^3*b^4)*d^2*e^3 + 5*(37*B*a^5*b^2 + 3*A*a^4*b^3)*d*e^4 - 15*(7*B*a^6*b + 3*A
*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (17*B*a*b^6 + 3*A*b^7)*d*e^4 + (7*B*a^2*b^5 + 3*A*a*b^6)*e^5)*x^4 + 10*
(118*B*b^7*d^3*e^2 - 3*(99*B*a*b^6 - A*b^7)*d^2*e^3 + 6*(43*B*a^2*b^5 - 4*A*a*b^6)*d*e^4 - (79*B*a^3*b^4 - 21*
A*a^2*b^5)*e^5)*x^3 + 2*(680*B*b^7*d^4*e - 2*(661*B*a*b^6 - 186*A*b^7)*d^3*e^2 + 3*(97*B*a^2*b^5 - 357*A*a*b^6
)*d^2*e^3 + (799*B*a^3*b^4 + 891*A*a^2*b^5)*d*e^4 - 64*(7*B*a^4*b^3 + 3*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5
- 8*(43*B*a*b^6 - 63*A*b^7)*d^4*e + 2*(B*a^2*b^5 - 636*A*a*b^6)*d^3*e^2 - 3*(29*B*a^3*b^4 - 279*A*a^2*b^5)*d^
2*e^3 + 2*(217*B*a^4*b^3 + 18*A*a^3*b^4)*d*e^4 - 35*(7*B*a^5*b^2 + 3*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^
8*d^3 - 3*a^6*b^7*d^2*e + 3*a^7*b^6*d*e^2 - a^8*b^5*e^3 + (b^13*d^3 - 3*a*b^12*d^2*e + 3*a^2*b^11*d*e^2 - a^3*
b^10*e^3)*x^5 + 5*(a*b^12*d^3 - 3*a^2*b^11*d^2*e + 3*a^3*b^10*d*e^2 - a^4*b^9*e^3)*x^4 + 10*(a^2*b^11*d^3 - 3*
a^3*b^10*d^2*e + 3*a^4*b^9*d*e^2 - a^5*b^8*e^3)*x^3 + 10*(a^3*b^10*d^3 - 3*a^4*b^9*d^2*e + 3*a^5*b^8*d*e^2 - a
^6*b^7*e^3)*x^2 + 5*(a^4*b^9*d^3 - 3*a^5*b^8*d^2*e + 3*a^6*b^7*d*e^2 - a^7*b^6*e^3)*x), -1/1920*(15*(10*B*a^5*
b*d*e^4 - (7*B*a^6 + 3*A*a^5*b)*e^5 + (10*B*b^6*d*e^4 - (7*B*a*b^5 + 3*A*b^6)*e^5)*x^5 + 5*(10*B*a*b^5*d*e^4 -
(7*B*a^2*b^4 + 3*A*a*b^5)*e^5)*x^4 + 10*(10*B*a^2*b^4*d*e^4 - (7*B*a^3*b^3 + 3*A*a^2*b^4)*e^5)*x^3 + 10*(10*B
*a^3*b^3*d*e^4 - (7*B*a^4*b^2 + 3*A*a^3*b^3)*e^5)*x^2 + 5*(10*B*a^4*b^2*d*e^4 - (7*B*a^5*b + 3*A*a^4*b^2)*e^5)
*x)*sqrt(-b^2*d + a*b*e)*arctan(sqrt(-b^2*d + a*b*e)*sqrt(e*x + d)/(b*e*x + b*d)) + (96*(B*a*b^6 + 4*A*b^7)*d^
5 - 16*(8*B*a^2*b^5 + 57*A*a*b^6)*d^4*e - 12*(B*a^3*b^4 - 46*A*a^2*b^5)*d^3*e^2 - 6*(6*B*a^4*b^3 - A*a^3*b^4)*
d^2*e^3 + 5*(37*B*a^5*b^2 + 3*A*a^4*b^3)*d*e^4 - 15*(7*B*a^6*b + 3*A*a^5*b^2)*e^5 + 15*(10*B*b^7*d^2*e^3 - (17
*B*a*b^6 + 3*A*b^7)*d*e^4 + (7*B*a^2*b^5 + 3*A*a*b^6)*e^5)*x^4 + 10*(118*B*b^7*d^3*e^2 - 3*(99*B*a*b^6 - A*b^7
)*d^2*e^3 + 6*(43*B*a^2*b^5 - 4*A*a*b^6)*d*e^4 - (79*B*a^3*b^4 - 21*A*a^2*b^5)*e^5)*x^3 + 2*(680*B*b^7*d^4*e -
2*(661*B*a*b^6 - 186*A*b^7)*d^3*e^2 + 3*(97*B*a^2*b^5 - 357*A*a*b^6)*d^2*e^3 + (799*B*a^3*b^4 + 891*A*a^2*b^5
)*d*e^4 - 64*(7*B*a^4*b^3 + 3*A*a^3*b^4)*e^5)*x^2 + 2*(240*B*b^7*d^5 - 8*(43*B*a*b^6 - 63*A*b^7)*d^4*e + 2*(B*
a^2*b^5 - 636*A*a*b^6)*d^3*e^2 - 3*(29*B*a^3*b^4 - 279*A*a^2*b^5)*d^2*e^3 + 2*(217*B*a^4*b^3 + 18*A*a^3*b^4)*d
*e^4 - 35*(7*B*a^5*b^2 + 3*A*a^4*b^3)*e^5)*x)*sqrt(e*x + d))/(a^5*b^8*d^3 - 3*a^6*b^7*d^2*e + 3*a^7*b^6*d*e^2
- a^8*b^5*e^3 + (b^13*d^3 - 3*a*b^12*d^2*e + 3*a^2*b^11*d*e^2 - a^3*b^10*e^3)*x^5 + 5*(a*b^12*d^3 - 3*a^2*b^11
*d^2*e + 3*a^3*b^10*d*e^2 - a^4*b^9*e^3)*x^4 + 10*(a^2*b^11*d^3 - 3*a^3*b^10*d^2*e + 3*a^4*b^9*d*e^2 - a^5*b^8
*e^3)*x^3 + 10*(a^3*b^10*d^3 - 3*a^4*b^9*d^2*e + 3*a^5*b^8*d*e^2 - a^6*b^7*e^3)*x^2 + 5*(a^4*b^9*d^3 - 3*a^5*b
^8*d^2*e + 3*a^6*b^7*d*e^2 - a^7*b^6*e^3)*x)]

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giac [B]  time = 0.30, size = 805, normalized size = 2.57 \begin {gather*} -\frac {{\left (10 \, B b d e^{4} - 7 \, B a e^{5} - 3 \, A b e^{5}\right )} \arctan \left (\frac {\sqrt {x e + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {150 \, {\left (x e + d\right )}^{\frac {9}{2}} B b^{5} d e^{4} + 580 \, {\left (x e + d\right )}^{\frac {7}{2}} B b^{5} d^{2} e^{4} - 1280 \, {\left (x e + d\right )}^{\frac {5}{2}} B b^{5} d^{3} e^{4} + 700 \, {\left (x e + d\right )}^{\frac {3}{2}} B b^{5} d^{4} e^{4} - 150 \, \sqrt {x e + d} B b^{5} d^{5} e^{4} - 105 \, {\left (x e + d\right )}^{\frac {9}{2}} B a b^{4} e^{5} - 45 \, {\left (x e + d\right )}^{\frac {9}{2}} A b^{5} e^{5} - 1370 \, {\left (x e + d\right )}^{\frac {7}{2}} B a b^{4} d e^{5} + 210 \, {\left (x e + d\right )}^{\frac {7}{2}} A b^{5} d e^{5} + 3456 \, {\left (x e + d\right )}^{\frac {5}{2}} B a b^{4} d^{2} e^{5} + 384 \, {\left (x e + d\right )}^{\frac {5}{2}} A b^{5} d^{2} e^{5} - 2590 \, {\left (x e + d\right )}^{\frac {3}{2}} B a b^{4} d^{3} e^{5} - 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A b^{5} d^{3} e^{5} + 705 \, \sqrt {x e + d} B a b^{4} d^{4} e^{5} + 45 \, \sqrt {x e + d} A b^{5} d^{4} e^{5} + 790 \, {\left (x e + d\right )}^{\frac {7}{2}} B a^{2} b^{3} e^{6} - 210 \, {\left (x e + d\right )}^{\frac {7}{2}} A a b^{4} e^{6} - 3072 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} b^{3} d e^{6} - 768 \, {\left (x e + d\right )}^{\frac {5}{2}} A a b^{4} d e^{6} + 3570 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} b^{3} d^{2} e^{6} + 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A a b^{4} d^{2} e^{6} - 1320 \, \sqrt {x e + d} B a^{2} b^{3} d^{3} e^{6} - 180 \, \sqrt {x e + d} A a b^{4} d^{3} e^{6} + 896 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{3} b^{2} e^{7} + 384 \, {\left (x e + d\right )}^{\frac {5}{2}} A a^{2} b^{3} e^{7} - 2170 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{3} b^{2} d e^{7} - 630 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} b^{3} d e^{7} + 1230 \, \sqrt {x e + d} B a^{3} b^{2} d^{2} e^{7} + 270 \, \sqrt {x e + d} A a^{2} b^{3} d^{2} e^{7} + 490 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{4} b e^{8} + 210 \, {\left (x e + d\right )}^{\frac {3}{2}} A a^{3} b^{2} e^{8} - 570 \, \sqrt {x e + d} B a^{4} b d e^{8} - 180 \, \sqrt {x e + d} A a^{3} b^{2} d e^{8} + 105 \, \sqrt {x e + d} B a^{5} e^{9} + 45 \, \sqrt {x e + d} A a^{4} b e^{9}}{1920 \, {\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} {\left ({\left (x e + d\right )} b - b d + a e\right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")

[Out]

-1/128*(10*B*b*d*e^4 - 7*B*a*e^5 - 3*A*b*e^5)*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))/((b^6*d^2 - 2*a*b^5
*d*e + a^2*b^4*e^2)*sqrt(-b^2*d + a*b*e)) - 1/1920*(150*(x*e + d)^(9/2)*B*b^5*d*e^4 + 580*(x*e + d)^(7/2)*B*b^
5*d^2*e^4 - 1280*(x*e + d)^(5/2)*B*b^5*d^3*e^4 + 700*(x*e + d)^(3/2)*B*b^5*d^4*e^4 - 150*sqrt(x*e + d)*B*b^5*d
^5*e^4 - 105*(x*e + d)^(9/2)*B*a*b^4*e^5 - 45*(x*e + d)^(9/2)*A*b^5*e^5 - 1370*(x*e + d)^(7/2)*B*a*b^4*d*e^5 +
210*(x*e + d)^(7/2)*A*b^5*d*e^5 + 3456*(x*e + d)^(5/2)*B*a*b^4*d^2*e^5 + 384*(x*e + d)^(5/2)*A*b^5*d^2*e^5 -
2590*(x*e + d)^(3/2)*B*a*b^4*d^3*e^5 - 210*(x*e + d)^(3/2)*A*b^5*d^3*e^5 + 705*sqrt(x*e + d)*B*a*b^4*d^4*e^5 +
45*sqrt(x*e + d)*A*b^5*d^4*e^5 + 790*(x*e + d)^(7/2)*B*a^2*b^3*e^6 - 210*(x*e + d)^(7/2)*A*a*b^4*e^6 - 3072*(
x*e + d)^(5/2)*B*a^2*b^3*d*e^6 - 768*(x*e + d)^(5/2)*A*a*b^4*d*e^6 + 3570*(x*e + d)^(3/2)*B*a^2*b^3*d^2*e^6 +
630*(x*e + d)^(3/2)*A*a*b^4*d^2*e^6 - 1320*sqrt(x*e + d)*B*a^2*b^3*d^3*e^6 - 180*sqrt(x*e + d)*A*a*b^4*d^3*e^6
+ 896*(x*e + d)^(5/2)*B*a^3*b^2*e^7 + 384*(x*e + d)^(5/2)*A*a^2*b^3*e^7 - 2170*(x*e + d)^(3/2)*B*a^3*b^2*d*e^
7 - 630*(x*e + d)^(3/2)*A*a^2*b^3*d*e^7 + 1230*sqrt(x*e + d)*B*a^3*b^2*d^2*e^7 + 270*sqrt(x*e + d)*A*a^2*b^3*d
^2*e^7 + 490*(x*e + d)^(3/2)*B*a^4*b*e^8 + 210*(x*e + d)^(3/2)*A*a^3*b^2*e^8 - 570*sqrt(x*e + d)*B*a^4*b*d*e^8
- 180*sqrt(x*e + d)*A*a^3*b^2*d*e^8 + 105*sqrt(x*e + d)*B*a^5*e^9 + 45*sqrt(x*e + d)*A*a^4*b*e^9)/((b^6*d^2 -
2*a*b^5*d*e + a^2*b^4*e^2)*((x*e + d)*b - b*d + a*e)^5)

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maple [B]  time = 0.08, size = 872, normalized size = 2.79 \begin {gather*} -\frac {3 \sqrt {e x +d}\, A \,a^{2} e^{7}}{128 \left (b e x +a e \right )^{5} b^{3}}+\frac {3 \sqrt {e x +d}\, A a d \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {3 \left (e x +d \right )^{\frac {9}{2}} A b \,e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {3 \sqrt {e x +d}\, A \,d^{2} e^{5}}{128 \left (b e x +a e \right )^{5} b}-\frac {7 \sqrt {e x +d}\, B \,a^{3} e^{7}}{128 \left (b e x +a e \right )^{5} b^{4}}+\frac {3 \sqrt {e x +d}\, B \,a^{2} d \,e^{6}}{16 \left (b e x +a e \right )^{5} b^{3}}-\frac {27 \sqrt {e x +d}\, B a \,d^{2} e^{5}}{128 \left (b e x +a e \right )^{5} b^{2}}+\frac {7 \left (e x +d \right )^{\frac {9}{2}} B a \,e^{5}}{128 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}-\frac {5 \left (e x +d \right )^{\frac {9}{2}} B b d \,e^{4}}{64 \left (b e x +a e \right )^{5} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {5 \sqrt {e x +d}\, B \,d^{3} e^{4}}{64 \left (b e x +a e \right )^{5} b}-\frac {7 \left (e x +d \right )^{\frac {3}{2}} A a \,e^{6}}{64 \left (b e x +a e \right )^{5} b^{2}}+\frac {7 \left (e x +d \right )^{\frac {3}{2}} A d \,e^{5}}{64 \left (b e x +a e \right )^{5} b}+\frac {7 \left (e x +d \right )^{\frac {7}{2}} A \,e^{5}}{64 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {49 \left (e x +d \right )^{\frac {3}{2}} B \,a^{2} e^{6}}{192 \left (b e x +a e \right )^{5} b^{3}}-\frac {79 \left (e x +d \right )^{\frac {7}{2}} B a \,e^{5}}{192 \left (b e x +a e \right )^{5} \left (a e -b d \right ) b}+\frac {119 \left (e x +d \right )^{\frac {3}{2}} B a d \,e^{5}}{192 \left (b e x +a e \right )^{5} b^{2}}-\frac {35 \left (e x +d \right )^{\frac {3}{2}} B \,d^{2} e^{4}}{96 \left (b e x +a e \right )^{5} b}+\frac {29 \left (e x +d \right )^{\frac {7}{2}} B d \,e^{4}}{96 \left (b e x +a e \right )^{5} \left (a e -b d \right )}-\frac {\left (e x +d \right )^{\frac {5}{2}} A \,e^{5}}{5 \left (b e x +a e \right )^{5} b}-\frac {7 \left (e x +d \right )^{\frac {5}{2}} B a \,e^{5}}{15 \left (b e x +a e \right )^{5} b^{2}}+\frac {2 \left (e x +d \right )^{\frac {5}{2}} B d \,e^{4}}{3 \left (b e x +a e \right )^{5} b}+\frac {3 A \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{3}}+\frac {7 B a \,e^{5} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{4}}-\frac {5 B d \,e^{4} \arctan \left (\frac {\sqrt {e x +d}\, b}{\sqrt {\left (a e -b d \right ) b}}\right )}{64 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}\, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

3/128*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*A*b+7/128*e^5/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d
*e+b^2*d^2)*(e*x+d)^(9/2)*a*B-5/64*e^4/(b*e*x+a*e)^5/(a^2*e^2-2*a*b*d*e+b^2*d^2)*(e*x+d)^(9/2)*B*b*d+7/64*e^5/
(b*e*x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)*A-79/192*e^5/(b*e*x+a*e)^5/b/(a*e-b*d)*(e*x+d)^(7/2)*a*B+29/96*e^4/(b*e*
x+a*e)^5/(a*e-b*d)*(e*x+d)^(7/2)*B*d-1/5*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*A-7/15*e^5/(b*e*x+a*e)^5/b^2*(e*x+d
)^(5/2)*a*B+2/3*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(5/2)*B*d-7/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(3/2)*A*a+7/64*e^5/(b
*e*x+a*e)^5/b*(e*x+d)^(3/2)*A*d-49/192*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(3/2)*B*a^2+119/192*e^5/(b*e*x+a*e)^5/b^2
*(e*x+d)^(3/2)*B*a*d-35/96*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(3/2)*B*d^2-3/128*e^7/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*A
*a^2+3/64*e^6/(b*e*x+a*e)^5/b^2*(e*x+d)^(1/2)*A*a*d-3/128*e^5/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*A*d^2-7/128*e^7/(b
*e*x+a*e)^5/b^4*(e*x+d)^(1/2)*B*a^3+3/16*e^6/(b*e*x+a*e)^5/b^3*(e*x+d)^(1/2)*B*a^2*d-27/128*e^5/(b*e*x+a*e)^5/
b^2*(e*x+d)^(1/2)*B*a*d^2+5/64*e^4/(b*e*x+a*e)^5/b*(e*x+d)^(1/2)*B*d^3+3/128*e^5/b^3/(a^2*e^2-2*a*b*d*e+b^2*d^
2)/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*A+7/128*e^5/b^4/(a^2*e^2-2*a*b*d*e+b^2*d^2)
/((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*a*B-5/64*e^4/b^3/(a^2*e^2-2*a*b*d*e+b^2*d^2)/
((a*e-b*d)*b)^(1/2)*arctan((e*x+d)^(1/2)/((a*e-b*d)*b)^(1/2)*b)*B*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(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 = 0.35, size = 572, normalized size = 1.83 \begin {gather*} \frac {e^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^4\,\sqrt {d+e\,x}\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{\sqrt {a\,e-b\,d}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}\right )\,\left (3\,A\,b\,e+7\,B\,a\,e-10\,B\,b\,d\right )}{128\,b^{9/2}\,{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{15\,b^2}-\frac {{\left (d+e\,x\right )}^{9/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,{\left (a\,e-b\,d\right )}^2}+\frac {7\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{192\,b^3}-\frac {{\left (d+e\,x\right )}^{7/2}\,\left (21\,A\,b\,e^5-79\,B\,a\,e^5+58\,B\,b\,d\,e^4\right )}{192\,b\,\left (a\,e-b\,d\right )}+\frac {\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )\,\left (3\,A\,b\,e^5+7\,B\,a\,e^5-10\,B\,b\,d\,e^4\right )}{128\,b^4}}{\left (d+e\,x\right )\,\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 )-{\left (d+e\,x\right )}^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 )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(5/2))/(a^2 + b^2*x^2 + 2*a*b*x)^3,x)

[Out]

(e^4*atan((b^(1/2)*e^4*(d + e*x)^(1/2)*(3*A*b*e + 7*B*a*e - 10*B*b*d))/((a*e - b*d)^(1/2)*(3*A*b*e^5 + 7*B*a*e
^5 - 10*B*b*d*e^4)))*(3*A*b*e + 7*B*a*e - 10*B*b*d))/(128*b^(9/2)*(a*e - b*d)^(5/2)) - (((d + e*x)^(5/2)*(3*A*
b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(15*b^2) - ((d + e*x)^(9/2)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(128*(a
*e - b*d)^2) + (7*(a*e - b*d)*(d + e*x)^(3/2)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(192*b^3) - ((d + e*x)^(
7/2)*(21*A*b*e^5 - 79*B*a*e^5 + 58*B*b*d*e^4))/(192*b*(a*e - b*d)) + ((d + e*x)^(1/2)*(a^2*e^2 + b^2*d^2 - 2*a
*b*d*e)*(3*A*b*e^5 + 7*B*a*e^5 - 10*B*b*d*e^4))/(128*b^4))/((d + e*x)*(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) - (d + e*x)^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) + b^5*(d + e*x)^5 - (5*b^5*d - 5*a*b^4*e)*(d + e*x)^4 + a^5*e^5 - b^5*d^5 + (d + e*x)^3*(10*b^5*d
^2 + 10*a^2*b^3*e^2 - 20*a*b^4*d*e) - 10*a^2*b^3*d^3*e^2 + 10*a^3*b^2*d^2*e^3 + 5*a*b^4*d^4*e - 5*a^4*b*d*e^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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