3.16.36 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=438 \[ \frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{6 e^7 (a+b x) (d+e x)^6}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x) (d+e x)^7}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5} \]

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Rubi [A]  time = 0.32, antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {770, 77} \begin {gather*} \frac {b^4 \sqrt {a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{6 e^7 (a+b x) (d+e x)^6}-\frac {5 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x) (d+e x)^7}+\frac {5 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-((b*d - a*e)^5*(B*d - A*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^7*(a + b*x)*(d + e*x)^11) + ((b*d - a*e)^4*(6
*b*B*d - 5*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(10*e^7*(a + b*x)*(d + e*x)^10) - (5*b*(b*d - a*e)^3*
(3*b*B*d - 2*A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*e^7*(a + b*x)*(d + e*x)^9) + (5*b^2*(b*d - a*e)^
2*(2*b*B*d - A*b*e - a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*e^7*(a + b*x)*(d + e*x)^8) - (5*b^3*(b*d - a*e)*
(3*b*B*d - A*b*e - 2*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^7*(a + b*x)*(d + e*x)^7) + (b^4*(6*b*B*d - A*b
*e - 5*a*B*e)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(6*e^7*(a + b*x)*(d + e*x)^6) - (b^5*B*Sqrt[a^2 + 2*a*b*x + b^2*x
^2])/(5*e^7*(a + b*x)*(d + e*x)^5)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{12}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{(d+e x)^{12}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (-B d+A e)}{e^6 (d+e x)^{12}}+\frac {b^5 (b d-a e)^4 (-6 b B d+5 A b e+a B e)}{e^6 (d+e x)^{11}}-\frac {5 b^6 (b d-a e)^3 (-3 b B d+2 A b e+a B e)}{e^6 (d+e x)^{10}}+\frac {10 b^7 (b d-a e)^2 (-2 b B d+A b e+a B e)}{e^6 (d+e x)^9}-\frac {5 b^8 (b d-a e) (-3 b B d+A b e+2 a B e)}{e^6 (d+e x)^8}+\frac {b^9 (-6 b B d+A b e+5 a B e)}{e^6 (d+e x)^7}+\frac {b^{10} B}{e^6 (d+e x)^6}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=-\frac {(b d-a e)^5 (B d-A e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}+\frac {(b d-a e)^4 (6 b B d-5 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}-\frac {5 b (b d-a e)^3 (3 b B d-2 A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x) (d+e x)^9}+\frac {5 b^2 (b d-a e)^2 (2 b B d-A b e-a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^8}-\frac {5 b^3 (b d-a e) (3 b B d-A b e-2 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^7}+\frac {b^4 (6 b B d-A b e-5 a B e) \sqrt {a^2+2 a b x+b^2 x^2}}{6 e^7 (a+b x) (d+e x)^6}-\frac {b^5 B \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 471, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (126 a^5 e^5 (10 A e+B (d+11 e x))+70 a^4 b e^4 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+35 a^3 b^2 e^3 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )\right )+15 a^2 b^3 e^2 \left (7 A e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+5 a b^4 e \left (6 A e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+b^5 \left (5 A e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )\right )}{13860 e^7 (a+b x) (d+e x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

-1/13860*(Sqrt[(a + b*x)^2]*(126*a^5*e^5*(10*A*e + B*(d + 11*e*x)) + 70*a^4*b*e^4*(9*A*e*(d + 11*e*x) + 2*B*(d
^2 + 11*d*e*x + 55*e^2*x^2)) + 35*a^3*b^2*e^3*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 5
5*d*e^2*x^2 + 165*e^3*x^3)) + 15*a^2*b^3*e^2*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4
+ 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + 5*a*b^4*e*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e
^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x
^4 + 462*e^5*x^5)) + b^5*(5*A*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5
*x^5) + 6*B*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x
^6))))/(e^7*(a + b*x)*(d + e*x)^11)

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IntegrateAlgebraic [F]  time = 180.10, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^12,x]

[Out]

$Aborted

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fricas [A]  time = 0.45, size = 673, normalized size = 1.54 \begin {gather*} -\frac {2772 \, B b^{5} e^{6} x^{6} + 6 \, B b^{5} d^{6} + 1260 \, A a^{5} e^{6} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{5} e + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{4} e^{2} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{3} e^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d^{2} e^{4} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} d e^{5} + 462 \, {\left (6 \, B b^{5} d e^{5} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} e^{6}\right )} x^{5} + 330 \, {\left (6 \, B b^{5} d^{2} e^{4} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d e^{5} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} e^{6}\right )} x^{4} + 165 \, {\left (6 \, B b^{5} d^{3} e^{3} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{2} e^{4} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d e^{5} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} e^{6}\right )} x^{3} + 55 \, {\left (6 \, B b^{5} d^{4} e^{2} + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{3} e^{3} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{2} e^{4} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d e^{5} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} e^{6}\right )} x^{2} + 11 \, {\left (6 \, B b^{5} d^{5} e + 5 \, {\left (5 \, B a b^{4} + A b^{5}\right )} d^{4} e^{2} + 30 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} d^{3} e^{3} + 105 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} d^{2} e^{4} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} d e^{5} + 126 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} e^{6}\right )} x}{13860 \, {\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/13860*(2772*B*b^5*e^6*x^6 + 6*B*b^5*d^6 + 1260*A*a^5*e^6 + 5*(5*B*a*b^4 + A*b^5)*d^5*e + 30*(2*B*a^2*b^3 +
A*a*b^4)*d^4*e^2 + 105*(B*a^3*b^2 + A*a^2*b^3)*d^3*e^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*d^2*e^4 + 126*(B*a^5 + 5*
A*a^4*b)*d*e^5 + 462*(6*B*b^5*d*e^5 + 5*(5*B*a*b^4 + A*b^5)*e^6)*x^5 + 330*(6*B*b^5*d^2*e^4 + 5*(5*B*a*b^4 + A
*b^5)*d*e^5 + 30*(2*B*a^2*b^3 + A*a*b^4)*e^6)*x^4 + 165*(6*B*b^5*d^3*e^3 + 5*(5*B*a*b^4 + A*b^5)*d^2*e^4 + 30*
(2*B*a^2*b^3 + A*a*b^4)*d*e^5 + 105*(B*a^3*b^2 + A*a^2*b^3)*e^6)*x^3 + 55*(6*B*b^5*d^4*e^2 + 5*(5*B*a*b^4 + A*
b^5)*d^3*e^3 + 30*(2*B*a^2*b^3 + A*a*b^4)*d^2*e^4 + 105*(B*a^3*b^2 + A*a^2*b^3)*d*e^5 + 140*(B*a^4*b + 2*A*a^3
*b^2)*e^6)*x^2 + 11*(6*B*b^5*d^5*e + 5*(5*B*a*b^4 + A*b^5)*d^4*e^2 + 30*(2*B*a^2*b^3 + A*a*b^4)*d^3*e^3 + 105*
(B*a^3*b^2 + A*a^2*b^3)*d^2*e^4 + 140*(B*a^4*b + 2*A*a^3*b^2)*d*e^5 + 126*(B*a^5 + 5*A*a^4*b)*e^6)*x)/(e^18*x^
11 + 11*d*e^17*x^10 + 55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d^6*e^12*
x^5 + 330*d^7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8*x + d^11*e^7)

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giac [B]  time = 0.36, size = 919, normalized size = 2.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/13860*(2772*B*b^5*x^6*e^6*sgn(b*x + a) + 2772*B*b^5*d*x^5*e^5*sgn(b*x + a) + 1980*B*b^5*d^2*x^4*e^4*sgn(b*x
+ a) + 990*B*b^5*d^3*x^3*e^3*sgn(b*x + a) + 330*B*b^5*d^4*x^2*e^2*sgn(b*x + a) + 66*B*b^5*d^5*x*e*sgn(b*x + a
) + 6*B*b^5*d^6*sgn(b*x + a) + 11550*B*a*b^4*x^5*e^6*sgn(b*x + a) + 2310*A*b^5*x^5*e^6*sgn(b*x + a) + 8250*B*a
*b^4*d*x^4*e^5*sgn(b*x + a) + 1650*A*b^5*d*x^4*e^5*sgn(b*x + a) + 4125*B*a*b^4*d^2*x^3*e^4*sgn(b*x + a) + 825*
A*b^5*d^2*x^3*e^4*sgn(b*x + a) + 1375*B*a*b^4*d^3*x^2*e^3*sgn(b*x + a) + 275*A*b^5*d^3*x^2*e^3*sgn(b*x + a) +
275*B*a*b^4*d^4*x*e^2*sgn(b*x + a) + 55*A*b^5*d^4*x*e^2*sgn(b*x + a) + 25*B*a*b^4*d^5*e*sgn(b*x + a) + 5*A*b^5
*d^5*e*sgn(b*x + a) + 19800*B*a^2*b^3*x^4*e^6*sgn(b*x + a) + 9900*A*a*b^4*x^4*e^6*sgn(b*x + a) + 9900*B*a^2*b^
3*d*x^3*e^5*sgn(b*x + a) + 4950*A*a*b^4*d*x^3*e^5*sgn(b*x + a) + 3300*B*a^2*b^3*d^2*x^2*e^4*sgn(b*x + a) + 165
0*A*a*b^4*d^2*x^2*e^4*sgn(b*x + a) + 660*B*a^2*b^3*d^3*x*e^3*sgn(b*x + a) + 330*A*a*b^4*d^3*x*e^3*sgn(b*x + a)
+ 60*B*a^2*b^3*d^4*e^2*sgn(b*x + a) + 30*A*a*b^4*d^4*e^2*sgn(b*x + a) + 17325*B*a^3*b^2*x^3*e^6*sgn(b*x + a)
+ 17325*A*a^2*b^3*x^3*e^6*sgn(b*x + a) + 5775*B*a^3*b^2*d*x^2*e^5*sgn(b*x + a) + 5775*A*a^2*b^3*d*x^2*e^5*sgn(
b*x + a) + 1155*B*a^3*b^2*d^2*x*e^4*sgn(b*x + a) + 1155*A*a^2*b^3*d^2*x*e^4*sgn(b*x + a) + 105*B*a^3*b^2*d^3*e
^3*sgn(b*x + a) + 105*A*a^2*b^3*d^3*e^3*sgn(b*x + a) + 7700*B*a^4*b*x^2*e^6*sgn(b*x + a) + 15400*A*a^3*b^2*x^2
*e^6*sgn(b*x + a) + 1540*B*a^4*b*d*x*e^5*sgn(b*x + a) + 3080*A*a^3*b^2*d*x*e^5*sgn(b*x + a) + 140*B*a^4*b*d^2*
e^4*sgn(b*x + a) + 280*A*a^3*b^2*d^2*e^4*sgn(b*x + a) + 1386*B*a^5*x*e^6*sgn(b*x + a) + 6930*A*a^4*b*x*e^6*sgn
(b*x + a) + 126*B*a^5*d*e^5*sgn(b*x + a) + 630*A*a^4*b*d*e^5*sgn(b*x + a) + 1260*A*a^5*e^6*sgn(b*x + a))*e^(-7
)/(x*e + d)^11

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maple [A]  time = 0.05, size = 689, normalized size = 1.57 \begin {gather*} -\frac {\left (2772 B \,b^{5} e^{6} x^{6}+2310 A \,b^{5} e^{6} x^{5}+11550 B a \,b^{4} e^{6} x^{5}+2772 B \,b^{5} d \,e^{5} x^{5}+9900 A a \,b^{4} e^{6} x^{4}+1650 A \,b^{5} d \,e^{5} x^{4}+19800 B \,a^{2} b^{3} e^{6} x^{4}+8250 B a \,b^{4} d \,e^{5} x^{4}+1980 B \,b^{5} d^{2} e^{4} x^{4}+17325 A \,a^{2} b^{3} e^{6} x^{3}+4950 A a \,b^{4} d \,e^{5} x^{3}+825 A \,b^{5} d^{2} e^{4} x^{3}+17325 B \,a^{3} b^{2} e^{6} x^{3}+9900 B \,a^{2} b^{3} d \,e^{5} x^{3}+4125 B a \,b^{4} d^{2} e^{4} x^{3}+990 B \,b^{5} d^{3} e^{3} x^{3}+15400 A \,a^{3} b^{2} e^{6} x^{2}+5775 A \,a^{2} b^{3} d \,e^{5} x^{2}+1650 A a \,b^{4} d^{2} e^{4} x^{2}+275 A \,b^{5} d^{3} e^{3} x^{2}+7700 B \,a^{4} b \,e^{6} x^{2}+5775 B \,a^{3} b^{2} d \,e^{5} x^{2}+3300 B \,a^{2} b^{3} d^{2} e^{4} x^{2}+1375 B a \,b^{4} d^{3} e^{3} x^{2}+330 B \,b^{5} d^{4} e^{2} x^{2}+6930 A \,a^{4} b \,e^{6} x +3080 A \,a^{3} b^{2} d \,e^{5} x +1155 A \,a^{2} b^{3} d^{2} e^{4} x +330 A a \,b^{4} d^{3} e^{3} x +55 A \,b^{5} d^{4} e^{2} x +1386 B \,a^{5} e^{6} x +1540 B \,a^{4} b d \,e^{5} x +1155 B \,a^{3} b^{2} d^{2} e^{4} x +660 B \,a^{2} b^{3} d^{3} e^{3} x +275 B a \,b^{4} d^{4} e^{2} x +66 B \,b^{5} d^{5} e x +1260 A \,a^{5} e^{6}+630 A \,a^{4} b d \,e^{5}+280 A \,a^{3} b^{2} d^{2} e^{4}+105 A \,a^{2} b^{3} d^{3} e^{3}+30 A a \,b^{4} d^{4} e^{2}+5 A \,b^{5} d^{5} e +126 B \,a^{5} d \,e^{5}+140 B \,a^{4} b \,d^{2} e^{4}+105 B \,a^{3} b^{2} d^{3} e^{3}+60 B \,a^{2} b^{3} d^{4} e^{2}+25 B a \,b^{4} d^{5} e +6 B \,b^{5} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{13860 \left (e x +d \right )^{11} \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,x)

[Out]

-1/13860/e^7*(2772*B*b^5*e^6*x^6+2310*A*b^5*e^6*x^5+11550*B*a*b^4*e^6*x^5+2772*B*b^5*d*e^5*x^5+9900*A*a*b^4*e^
6*x^4+1650*A*b^5*d*e^5*x^4+19800*B*a^2*b^3*e^6*x^4+8250*B*a*b^4*d*e^5*x^4+1980*B*b^5*d^2*e^4*x^4+17325*A*a^2*b
^3*e^6*x^3+4950*A*a*b^4*d*e^5*x^3+825*A*b^5*d^2*e^4*x^3+17325*B*a^3*b^2*e^6*x^3+9900*B*a^2*b^3*d*e^5*x^3+4125*
B*a*b^4*d^2*e^4*x^3+990*B*b^5*d^3*e^3*x^3+15400*A*a^3*b^2*e^6*x^2+5775*A*a^2*b^3*d*e^5*x^2+1650*A*a*b^4*d^2*e^
4*x^2+275*A*b^5*d^3*e^3*x^2+7700*B*a^4*b*e^6*x^2+5775*B*a^3*b^2*d*e^5*x^2+3300*B*a^2*b^3*d^2*e^4*x^2+1375*B*a*
b^4*d^3*e^3*x^2+330*B*b^5*d^4*e^2*x^2+6930*A*a^4*b*e^6*x+3080*A*a^3*b^2*d*e^5*x+1155*A*a^2*b^3*d^2*e^4*x+330*A
*a*b^4*d^3*e^3*x+55*A*b^5*d^4*e^2*x+1386*B*a^5*e^6*x+1540*B*a^4*b*d*e^5*x+1155*B*a^3*b^2*d^2*e^4*x+660*B*a^2*b
^3*d^3*e^3*x+275*B*a*b^4*d^4*e^2*x+66*B*b^5*d^5*e*x+1260*A*a^5*e^6+630*A*a^4*b*d*e^5+280*A*a^3*b^2*d^2*e^4+105
*A*a^2*b^3*d^3*e^3+30*A*a*b^4*d^4*e^2+5*A*b^5*d^5*e+126*B*a^5*d*e^5+140*B*a^4*b*d^2*e^4+105*B*a^3*b^2*d^3*e^3+
60*B*a^2*b^3*d^4*e^2+25*B*a*b^4*d^5*e+6*B*b^5*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^11/(b*x+a)^5

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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)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^12,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 zero or nonzero?

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mupad [B]  time = 2.60, size = 1489, normalized size = 3.40

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^12,x)

[Out]

- (((10*B*b^5*d^2 - 4*A*b^5*d*e + 5*A*a*b^4*e^2 + 10*B*a^2*b^3*e^2 - 20*B*a*b^4*d*e)/(7*e^7) - (d*((b^4*(A*b*e
+ 5*B*a*e - 4*B*b*d))/(7*e^6) - (B*b^5*d)/(7*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^
7) - (((A*b^5*e - 5*B*b^5*d + 5*B*a*b^4*e)/(6*e^7) - (B*b^5*d)/(6*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a +
b*x)*(d + e*x)^6) - (((A*a^5)/(11*e) - (d*((B*a^5 + 5*A*a^4*b)/(11*e) + (d*((d*((d*((d*((A*b^5 + 5*B*a*b^4)/(
11*e) - (B*b^5*d)/(11*e^2)))/e - (5*a*b^3*(A*b + 2*B*a))/(11*e)))/e + (10*a^2*b^2*(A*b + B*a))/(11*e)))/e - (5
*a^3*b*(2*A*b + B*a))/(11*e)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^11) - (((6*A*b^5*d
^2*e - 10*B*b^5*d^3 + 10*A*a^2*b^3*e^3 + 10*B*a^3*b^2*e^3 - 30*B*a^2*b^3*d*e^2 - 15*A*a*b^4*d*e^2 + 30*B*a*b^4
*d^2*e)/(8*e^7) - (d*((5*A*a*b^4*e^3 - 3*A*b^5*d*e^2 + 6*B*b^5*d^2*e + 10*B*a^2*b^3*e^3 - 15*B*a*b^4*d*e^2)/(8
*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 3*B*b*d))/(8*e^5) - (B*b^5*d)/(8*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(
1/2))/((a + b*x)*(d + e*x)^8) - (((B*a^5*e^5 - B*b^5*d^5 + 5*A*a^4*b*e^5 + A*b^5*d^4*e - 5*A*a*b^4*d^3*e^2 - 1
0*A*a^3*b^2*d*e^4 + 10*A*a^2*b^3*d^2*e^3 - 10*B*a^2*b^3*d^3*e^2 + 10*B*a^3*b^2*d^2*e^3 + 5*B*a*b^4*d^4*e - 5*B
*a^4*b*d*e^4)/(10*e^7) - (d*((5*B*a^4*b*e^5 + B*b^5*d^4*e + 10*A*a^3*b^2*e^5 - A*b^5*d^3*e^2 + 5*A*a*b^4*d^2*e
^3 - 10*A*a^2*b^3*d*e^4 - 5*B*a*b^4*d^3*e^2 - 10*B*a^3*b^2*d*e^4 + 10*B*a^2*b^3*d^2*e^3)/(10*e^7) - (d*((10*A*
a^2*b^3*e^5 + 10*B*a^3*b^2*e^5 + A*b^5*d^2*e^3 - B*b^5*d^3*e^2 + 5*B*a*b^4*d^2*e^3 - 10*B*a^2*b^3*d*e^4 - 5*A*
a*b^4*d*e^4)/(10*e^7) - (d*((5*A*a*b^4*e^5 - A*b^5*d*e^4 + 10*B*a^2*b^3*e^5 + B*b^5*d^2*e^3 - 5*B*a*b^4*d*e^4)
/(10*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - B*b*d))/(10*e^3) - (B*b^5*d)/(10*e^3)))/e))/e))/e))/e)*(a^2 + b^2*x^2
+ 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^10) - (((5*B*b^5*d^4 + 5*B*a^4*b*e^4 - 4*A*b^5*d^3*e + 10*A*a^3*b^2*e^4
+ 15*A*a*b^4*d^2*e^2 - 20*A*a^2*b^3*d*e^3 - 20*B*a^3*b^2*d*e^3 + 30*B*a^2*b^3*d^2*e^2 - 20*B*a*b^4*d^3*e)/(9*
e^7) - (d*((10*A*a^2*b^3*e^4 - 4*B*b^5*d^3*e + 10*B*a^3*b^2*e^4 + 3*A*b^5*d^2*e^2 + 15*B*a*b^4*d^2*e^2 - 20*B*
a^2*b^3*d*e^3 - 10*A*a*b^4*d*e^3)/(9*e^7) - (d*((5*A*a*b^4*e^4 - 2*A*b^5*d*e^3 + 10*B*a^2*b^3*e^4 + 3*B*b^5*d^
2*e^2 - 10*B*a*b^4*d*e^3)/(9*e^7) - (d*((b^4*(A*b*e + 5*B*a*e - 2*B*b*d))/(9*e^4) - (B*b^5*d)/(9*e^4)))/e))/e)
)/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^9) - (B*b^5*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(5*e^7
*(a + b*x)*(d + e*x)^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**12,x)

[Out]

Exception raised: HeuristicGCDFailed

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