3.1756 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^{12}} \, dx\)

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

[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)

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Rubi [A]  time = 0.323483, antiderivative size = 438, normalized size of antiderivative = 1., 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} \[ \frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (-5 a B e-A b e+6 b B d)}{6 e^7 (a+b x) (d+e x)^6}-\frac{5 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-2 a B e-A b e+3 b B d)}{7 e^7 (a+b x) (d+e x)^7}+\frac{5 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-A b e+2 b B d)}{4 e^7 (a+b x) (d+e x)^8}-\frac{5 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (-a B e-2 A b e+3 b B d)}{9 e^7 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^4 (-a B e-5 A b e+6 b B d)}{10 e^7 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^5 (B d-A e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^5} \]

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 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]

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]))))

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*}

Mathematica [A]  time = 0.24623, size = 471, normalized size = 1.08 \[ -\frac{\sqrt{(a+b x)^2} \left (15 a^2 b^3 e^2 \left (7 A e \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )\right )+35 a^3 b^2 e^3 \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )\right )+70 a^4 b e^4 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+126 a^5 e^5 (10 A e+B (d+11 e x))+5 a b^4 e \left (6 A e \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )+5 B \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )\right )+b^5 \left (5 A e \left (55 d^3 e^2 x^2+165 d^2 e^3 x^3+11 d^4 e x+d^5+330 d e^4 x^4+462 e^5 x^5\right )+6 B \left (55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+11 d^5 e x+d^6+462 d e^5 x^5+462 e^6 x^6\right )\right )\right )}{13860 e^7 (a+b x) (d+e x)^{11}} \]

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]

-(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 + 55*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))))/(
13860*e^7*(a + b*x)*(d + e*x)^11)

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Maple [A]  time = 0.01, size = 689, normalized size = 1.6 \begin{align*} -{\frac{2772\,B{x}^{6}{b}^{5}{e}^{6}+2310\,A{x}^{5}{b}^{5}{e}^{6}+11550\,B{x}^{5}a{b}^{4}{e}^{6}+2772\,B{x}^{5}{b}^{5}d{e}^{5}+9900\,A{x}^{4}a{b}^{4}{e}^{6}+1650\,A{x}^{4}{b}^{5}d{e}^{5}+19800\,B{x}^{4}{a}^{2}{b}^{3}{e}^{6}+8250\,B{x}^{4}a{b}^{4}d{e}^{5}+1980\,B{x}^{4}{b}^{5}{d}^{2}{e}^{4}+17325\,A{x}^{3}{a}^{2}{b}^{3}{e}^{6}+4950\,A{x}^{3}a{b}^{4}d{e}^{5}+825\,A{x}^{3}{b}^{5}{d}^{2}{e}^{4}+17325\,B{x}^{3}{a}^{3}{b}^{2}{e}^{6}+9900\,B{x}^{3}{a}^{2}{b}^{3}d{e}^{5}+4125\,B{x}^{3}a{b}^{4}{d}^{2}{e}^{4}+990\,B{x}^{3}{b}^{5}{d}^{3}{e}^{3}+15400\,A{x}^{2}{a}^{3}{b}^{2}{e}^{6}+5775\,A{x}^{2}{a}^{2}{b}^{3}d{e}^{5}+1650\,A{x}^{2}a{b}^{4}{d}^{2}{e}^{4}+275\,A{x}^{2}{b}^{5}{d}^{3}{e}^{3}+7700\,B{x}^{2}{a}^{4}b{e}^{6}+5775\,B{x}^{2}{a}^{3}{b}^{2}d{e}^{5}+3300\,B{x}^{2}{a}^{2}{b}^{3}{d}^{2}{e}^{4}+1375\,B{x}^{2}a{b}^{4}{d}^{3}{e}^{3}+330\,B{x}^{2}{b}^{5}{d}^{4}{e}^{2}+6930\,Ax{a}^{4}b{e}^{6}+3080\,Ax{a}^{3}{b}^{2}d{e}^{5}+1155\,Ax{a}^{2}{b}^{3}{d}^{2}{e}^{4}+330\,Axa{b}^{4}{d}^{3}{e}^{3}+55\,Ax{b}^{5}{d}^{4}{e}^{2}+1386\,Bx{a}^{5}{e}^{6}+1540\,Bx{a}^{4}bd{e}^{5}+1155\,Bx{a}^{3}{b}^{2}{d}^{2}{e}^{4}+660\,Bx{a}^{2}{b}^{3}{d}^{3}{e}^{3}+275\,Bxa{b}^{4}{d}^{4}{e}^{2}+66\,Bx{b}^{5}{d}^{5}e+1260\,A{a}^{5}{e}^{6}+630\,Ad{e}^{5}{a}^{4}b+280\,A{a}^{3}{b}^{2}{d}^{2}{e}^{4}+105\,A{a}^{2}{b}^{3}{d}^{3}{e}^{3}+30\,Aa{b}^{4}{d}^{4}{e}^{2}+5\,A{b}^{5}{d}^{5}e+126\,Bd{e}^{5}{a}^{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\,Ba{b}^{4}{d}^{5}e+6\,B{b}^{5}{d}^{6}}{13860\,{e}^{7} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [A]  time = 1.66158, size = 1455, normalized size = 3.32 \begin{align*} -\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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.19607, size = 1241, normalized size = 2.83 \begin{align*} \text{result too large to display} \end{align*}

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