3.1737 \(\int \frac{(A+B x) (a^2+2 a b x+b^2 x^2)^{3/2}}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

[Out]

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

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Rubi [A]  time = 0.178097, antiderivative size = 298, 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} \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{8 e^5 (a+b x) (d+e x)^8}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x) (d+e x)^9}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{10 e^5 (a+b x) (d+e x)^{10}}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{11 e^5 (a+b x) (d+e x)^{11}}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

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

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

Mathematica [A]  time = 0.117189, size = 233, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (28 a^2 b e^2 \left (9 A e (d+11 e x)+2 B \left (d^2+11 d e x+55 e^2 x^2\right )\right )+84 a^3 e^3 (10 A e+B (d+11 e x))+7 a b^2 e \left (8 A e \left (d^2+11 d e x+55 e^2 x^2\right )+3 B \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )\right )+b^3 \left (7 A e \left (11 d^2 e x+d^3+55 d e^2 x^2+165 e^3 x^3\right )+4 B \left (55 d^2 e^2 x^2+11 d^3 e x+d^4+165 d e^3 x^3+330 e^4 x^4\right )\right )\right )}{9240 e^5 (a+b x) (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^12,x]

[Out]

-(Sqrt[(a + b*x)^2]*(84*a^3*e^3*(10*A*e + B*(d + 11*e*x)) + 28*a^2*b*e^2*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d
*e*x + 55*e^2*x^2)) + 7*a*b^2*e*(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)) + b^3*(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))))/(9240*e^5*(a + b*x)*(d + e*x)^11)

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Maple [A]  time = 0.01, size = 317, normalized size = 1.1 \begin{align*} -{\frac{1320\,B{x}^{4}{b}^{3}{e}^{4}+1155\,A{x}^{3}{b}^{3}{e}^{4}+3465\,B{x}^{3}a{b}^{2}{e}^{4}+660\,B{x}^{3}{b}^{3}d{e}^{3}+3080\,A{x}^{2}a{b}^{2}{e}^{4}+385\,A{x}^{2}{b}^{3}d{e}^{3}+3080\,B{x}^{2}{a}^{2}b{e}^{4}+1155\,B{x}^{2}a{b}^{2}d{e}^{3}+220\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+2772\,Ax{a}^{2}b{e}^{4}+616\,Axa{b}^{2}d{e}^{3}+77\,Ax{b}^{3}{d}^{2}{e}^{2}+924\,Bx{a}^{3}{e}^{4}+616\,Bx{a}^{2}bd{e}^{3}+231\,Bxa{b}^{2}{d}^{2}{e}^{2}+44\,Bx{b}^{3}{d}^{3}e+840\,A{a}^{3}{e}^{4}+252\,Ad{e}^{3}{a}^{2}b+56\,Aa{b}^{2}{d}^{2}{e}^{2}+7\,A{b}^{3}{d}^{3}e+84\,Bd{e}^{3}{a}^{3}+56\,B{a}^{2}b{d}^{2}{e}^{2}+21\,Ba{b}^{2}{d}^{3}e+4\,B{b}^{3}{d}^{4}}{9240\,{e}^{5} \left ( ex+d \right ) ^{11} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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)^(3/2)/(e*x+d)^12,x)

[Out]

-1/9240/e^5*(1320*B*b^3*e^4*x^4+1155*A*b^3*e^4*x^3+3465*B*a*b^2*e^4*x^3+660*B*b^3*d*e^3*x^3+3080*A*a*b^2*e^4*x
^2+385*A*b^3*d*e^3*x^2+3080*B*a^2*b*e^4*x^2+1155*B*a*b^2*d*e^3*x^2+220*B*b^3*d^2*e^2*x^2+2772*A*a^2*b*e^4*x+61
6*A*a*b^2*d*e^3*x+77*A*b^3*d^2*e^2*x+924*B*a^3*e^4*x+616*B*a^2*b*d*e^3*x+231*B*a*b^2*d^2*e^2*x+44*B*b^3*d^3*e*
x+840*A*a^3*e^4+252*A*a^2*b*d*e^3+56*A*a*b^2*d^2*e^2+7*A*b^3*d^3*e+84*B*a^3*d*e^3+56*B*a^2*b*d^2*e^2+21*B*a*b^
2*d^3*e+4*B*b^3*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^11/(b*x+a)^3

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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)^(3/2)/(e*x+d)^12,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.58606, size = 826, normalized size = 2.77 \begin{align*} -\frac{1320 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 840 \, A a^{3} e^{4} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 56 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 84 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 165 \,{\left (4 \, B b^{3} d e^{3} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 55 \,{\left (4 \, B b^{3} d^{2} e^{2} + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 56 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 11 \,{\left (4 \, B b^{3} d^{3} e + 7 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 56 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 84 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{9240 \,{\left (e^{16} x^{11} + 11 \, d e^{15} x^{10} + 55 \, d^{2} e^{14} x^{9} + 165 \, d^{3} e^{13} x^{8} + 330 \, d^{4} e^{12} x^{7} + 462 \, d^{5} e^{11} x^{6} + 462 \, d^{6} e^{10} x^{5} + 330 \, d^{7} e^{9} x^{4} + 165 \, d^{8} e^{8} x^{3} + 55 \, d^{9} e^{7} x^{2} + 11 \, d^{10} e^{6} x + d^{11} e^{5}\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)^(3/2)/(e*x+d)^12,x, algorithm="fricas")

[Out]

-1/9240*(1320*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 840*A*a^3*e^4 + 7*(3*B*a*b^2 + A*b^3)*d^3*e + 56*(B*a^2*b + A*a*b^
2)*d^2*e^2 + 84*(B*a^3 + 3*A*a^2*b)*d*e^3 + 165*(4*B*b^3*d*e^3 + 7*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 55*(4*B*b^3*
d^2*e^2 + 7*(3*B*a*b^2 + A*b^3)*d*e^3 + 56*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 11*(4*B*b^3*d^3*e + 7*(3*B*a*b^2 + A
*b^3)*d^2*e^2 + 56*(B*a^2*b + A*a*b^2)*d*e^3 + 84*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^16*x^11 + 11*d*e^15*x^10 + 55
*d^2*e^14*x^9 + 165*d^3*e^13*x^8 + 330*d^4*e^12*x^7 + 462*d^5*e^11*x^6 + 462*d^6*e^10*x^5 + 330*d^7*e^9*x^4 +
165*d^8*e^8*x^3 + 55*d^9*e^7*x^2 + 11*d^10*e^6*x + d^11*e^5)

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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)**(3/2)/(e*x+d)**12,x)

[Out]

Timed out

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Giac [A]  time = 1.15712, size = 576, normalized size = 1.93 \begin{align*} -\frac{{\left (1320 \, B b^{3} x^{4} e^{4} \mathrm{sgn}\left (b x + a\right ) + 660 \, B b^{3} d x^{3} e^{3} \mathrm{sgn}\left (b x + a\right ) + 220 \, B b^{3} d^{2} x^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 44 \, B b^{3} d^{3} x e \mathrm{sgn}\left (b x + a\right ) + 4 \, B b^{3} d^{4} \mathrm{sgn}\left (b x + a\right ) + 3465 \, B a b^{2} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 1155 \, A b^{3} x^{3} e^{4} \mathrm{sgn}\left (b x + a\right ) + 1155 \, B a b^{2} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 385 \, A b^{3} d x^{2} e^{3} \mathrm{sgn}\left (b x + a\right ) + 231 \, B a b^{2} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 77 \, A b^{3} d^{2} x e^{2} \mathrm{sgn}\left (b x + a\right ) + 21 \, B a b^{2} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 7 \, A b^{3} d^{3} e \mathrm{sgn}\left (b x + a\right ) + 3080 \, B a^{2} b x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 3080 \, A a b^{2} x^{2} e^{4} \mathrm{sgn}\left (b x + a\right ) + 616 \, B a^{2} b d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 616 \, A a b^{2} d x e^{3} \mathrm{sgn}\left (b x + a\right ) + 56 \, B a^{2} b d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 56 \, A a b^{2} d^{2} e^{2} \mathrm{sgn}\left (b x + a\right ) + 924 \, B a^{3} x e^{4} \mathrm{sgn}\left (b x + a\right ) + 2772 \, A a^{2} b x e^{4} \mathrm{sgn}\left (b x + a\right ) + 84 \, B a^{3} d e^{3} \mathrm{sgn}\left (b x + a\right ) + 252 \, A a^{2} b d e^{3} \mathrm{sgn}\left (b x + a\right ) + 840 \, A a^{3} e^{4} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-5\right )}}{9240 \,{\left (x e + d\right )}^{11}} \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)^(3/2)/(e*x+d)^12,x, algorithm="giac")

[Out]

-1/9240*(1320*B*b^3*x^4*e^4*sgn(b*x + a) + 660*B*b^3*d*x^3*e^3*sgn(b*x + a) + 220*B*b^3*d^2*x^2*e^2*sgn(b*x +
a) + 44*B*b^3*d^3*x*e*sgn(b*x + a) + 4*B*b^3*d^4*sgn(b*x + a) + 3465*B*a*b^2*x^3*e^4*sgn(b*x + a) + 1155*A*b^3
*x^3*e^4*sgn(b*x + a) + 1155*B*a*b^2*d*x^2*e^3*sgn(b*x + a) + 385*A*b^3*d*x^2*e^3*sgn(b*x + a) + 231*B*a*b^2*d
^2*x*e^2*sgn(b*x + a) + 77*A*b^3*d^2*x*e^2*sgn(b*x + a) + 21*B*a*b^2*d^3*e*sgn(b*x + a) + 7*A*b^3*d^3*e*sgn(b*
x + a) + 3080*B*a^2*b*x^2*e^4*sgn(b*x + a) + 3080*A*a*b^2*x^2*e^4*sgn(b*x + a) + 616*B*a^2*b*d*x*e^3*sgn(b*x +
a) + 616*A*a*b^2*d*x*e^3*sgn(b*x + a) + 56*B*a^2*b*d^2*e^2*sgn(b*x + a) + 56*A*a*b^2*d^2*e^2*sgn(b*x + a) + 9
24*B*a^3*x*e^4*sgn(b*x + a) + 2772*A*a^2*b*x*e^4*sgn(b*x + a) + 84*B*a^3*d*e^3*sgn(b*x + a) + 252*A*a^2*b*d*e^
3*sgn(b*x + a) + 840*A*a^3*e^4*sgn(b*x + a))*e^(-5)/(x*e + d)^11