3.1733 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=298 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{4 e^5 (a+b x) (d+e x)^4}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x) (d+e x)^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.631735, 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 \[ \frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (-3 a B e-A b e+4 b B d)}{4 e^5 (a+b x) (d+e x)^4}-\frac{3 b \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e) (-a B e-A b e+2 b B d)}{5 e^5 (a+b x) (d+e x)^5}+\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{6 e^5 (a+b x) (d+e x)^6}-\frac{\sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^3 (B d-A e)}{7 e^5 (a+b x) (d+e x)^7}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x) (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 48.4907, size = 248, normalized size = 0.83 \[ \frac{b \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}} \left (3 A b e - 7 B a e + 4 B b d\right )}{420 \left (d + e x\right )^{5} \left (a e - b d\right )^{4}} - \frac{b \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (3 A b e - 7 B a e + 4 B b d\right )}{168 e \left (d + e x\right )^{5} \left (a e - b d\right )^{3}} + \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}} \left (3 A b e - 7 B a e + 4 B b d\right )}{84 e \left (d + e x\right )^{6} \left (a e - b d\right )^{2}} - \frac{\left (2 a + 2 b x\right ) \left (A e - B d\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{14 e \left (d + e x\right )^{7} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**(3/2)/(e*x+d)**8,x)

[Out]

b*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)*(3*A*b*e - 7*B*a*e + 4*B*b*d)/(420*(d + e*
x)**5*(a*e - b*d)**4) - b*(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**(3/2)*(3*A
*b*e - 7*B*a*e + 4*B*b*d)/(168*e*(d + e*x)**5*(a*e - b*d)**3) + (2*a + 2*b*x)*(a
**2 + 2*a*b*x + b**2*x**2)**(3/2)*(3*A*b*e - 7*B*a*e + 4*B*b*d)/(84*e*(d + e*x)*
*6*(a*e - b*d)**2) - (2*a + 2*b*x)*(A*e - B*d)*(a**2 + 2*a*b*x + b**2*x**2)**(3/
2)/(14*e*(d + e*x)**7*(a*e - b*d))

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Mathematica [A]  time = 0.192077, size = 233, normalized size = 0.78 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^3 e^3 (6 A e+B (d+7 e x))+6 a^2 b e^2 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a b^2 e \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+b^3 \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )\right )}{420 e^5 (a+b x) (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2))/(d + e*x)^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(10*a^3*e^3*(6*A*e + B*(d + 7*e*x)) + 6*a^2*b*e^2*(5*A*e*(d
+ 7*e*x) + 2*B*(d^2 + 7*d*e*x + 21*e^2*x^2)) + 3*a*b^2*e*(4*A*e*(d^2 + 7*d*e*x +
21*e^2*x^2) + 3*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + b^3*(3*A*e*(
d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2
*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))))/(420*e^5*(a + b*x)*(d + e*x)^7)

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Maple [A]  time = 0.015, size = 317, normalized size = 1.1 \[ -{\frac{140\,B{x}^{4}{b}^{3}{e}^{4}+105\,A{x}^{3}{b}^{3}{e}^{4}+315\,B{x}^{3}a{b}^{2}{e}^{4}+140\,B{x}^{3}{b}^{3}d{e}^{3}+252\,A{x}^{2}a{b}^{2}{e}^{4}+63\,A{x}^{2}{b}^{3}d{e}^{3}+252\,B{x}^{2}{a}^{2}b{e}^{4}+189\,B{x}^{2}a{b}^{2}d{e}^{3}+84\,B{x}^{2}{b}^{3}{d}^{2}{e}^{2}+210\,Ax{a}^{2}b{e}^{4}+84\,Axa{b}^{2}d{e}^{3}+21\,Ax{b}^{3}{d}^{2}{e}^{2}+70\,Bx{a}^{3}{e}^{4}+84\,Bx{a}^{2}bd{e}^{3}+63\,Bxa{b}^{2}{d}^{2}{e}^{2}+28\,Bx{b}^{3}{d}^{3}e+60\,A{a}^{3}{e}^{4}+30\,Ad{e}^{3}{a}^{2}b+12\,Aa{b}^{2}{d}^{2}{e}^{2}+3\,A{b}^{3}{d}^{3}e+10\,Bd{e}^{3}{a}^{3}+12\,B{a}^{2}b{d}^{2}{e}^{2}+9\,Ba{b}^{2}{d}^{3}e+4\,B{b}^{3}{d}^{4}}{420\,{e}^{5} \left ( ex+d \right ) ^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^8,x)

[Out]

-1/420/e^5*(140*B*b^3*e^4*x^4+105*A*b^3*e^4*x^3+315*B*a*b^2*e^4*x^3+140*B*b^3*d*
e^3*x^3+252*A*a*b^2*e^4*x^2+63*A*b^3*d*e^3*x^2+252*B*a^2*b*e^4*x^2+189*B*a*b^2*d
*e^3*x^2+84*B*b^3*d^2*e^2*x^2+210*A*a^2*b*e^4*x+84*A*a*b^2*d*e^3*x+21*A*b^3*d^2*
e^2*x+70*B*a^3*e^4*x+84*B*a^2*b*d*e^3*x+63*B*a*b^2*d^2*e^2*x+28*B*b^3*d^3*e*x+60
*A*a^3*e^4+30*A*a^2*b*d*e^3+12*A*a*b^2*d^2*e^2+3*A*b^3*d^3*e+10*B*a^3*d*e^3+12*B
*a^2*b*d^2*e^2+9*B*a*b^2*d^3*e+4*B*b^3*d^4)*((b*x+a)^2)^(3/2)/(e*x+d)^7/(b*x+a)^
3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312525, size = 448, normalized size = 1.5 \[ -\frac{140 \, B b^{3} e^{4} x^{4} + 4 \, B b^{3} d^{4} + 60 \, A a^{3} e^{4} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 35 \,{\left (4 \, B b^{3} d e^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 21 \,{\left (4 \, B b^{3} d^{2} e^{2} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 7 \,{\left (4 \, B b^{3} d^{3} e + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 10 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{420 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/420*(140*B*b^3*e^4*x^4 + 4*B*b^3*d^4 + 60*A*a^3*e^4 + 3*(3*B*a*b^2 + A*b^3)*d
^3*e + 12*(B*a^2*b + A*a*b^2)*d^2*e^2 + 10*(B*a^3 + 3*A*a^2*b)*d*e^3 + 35*(4*B*b
^3*d*e^3 + 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 + 21*(4*B*b^3*d^2*e^2 + 3*(3*B*a*b^2 +
A*b^3)*d*e^3 + 12*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 7*(4*B*b^3*d^3*e + 3*(3*B*a*b^
2 + A*b^3)*d^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + 10*(B*a^3 + 3*A*a^2*b)*e^4)*
x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^8*x^3
+ 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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

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)**8,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.301351, size = 576, normalized size = 1.93 \[ -\frac{{\left (140 \, B b^{3} x^{4} e^{4}{\rm sign}\left (b x + a\right ) + 140 \, B b^{3} d x^{3} e^{3}{\rm sign}\left (b x + a\right ) + 84 \, B b^{3} d^{2} x^{2} e^{2}{\rm sign}\left (b x + a\right ) + 28 \, B b^{3} d^{3} x e{\rm sign}\left (b x + a\right ) + 4 \, B b^{3} d^{4}{\rm sign}\left (b x + a\right ) + 315 \, B a b^{2} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 105 \, A b^{3} x^{3} e^{4}{\rm sign}\left (b x + a\right ) + 189 \, B a b^{2} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 63 \, A b^{3} d x^{2} e^{3}{\rm sign}\left (b x + a\right ) + 63 \, B a b^{2} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 21 \, A b^{3} d^{2} x e^{2}{\rm sign}\left (b x + a\right ) + 9 \, B a b^{2} d^{3} e{\rm sign}\left (b x + a\right ) + 3 \, A b^{3} d^{3} e{\rm sign}\left (b x + a\right ) + 252 \, B a^{2} b x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 252 \, A a b^{2} x^{2} e^{4}{\rm sign}\left (b x + a\right ) + 84 \, B a^{2} b d x e^{3}{\rm sign}\left (b x + a\right ) + 84 \, A a b^{2} d x e^{3}{\rm sign}\left (b x + a\right ) + 12 \, B a^{2} b d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 12 \, A a b^{2} d^{2} e^{2}{\rm sign}\left (b x + a\right ) + 70 \, B a^{3} x e^{4}{\rm sign}\left (b x + a\right ) + 210 \, A a^{2} b x e^{4}{\rm sign}\left (b x + a\right ) + 10 \, B a^{3} d e^{3}{\rm sign}\left (b x + a\right ) + 30 \, A a^{2} b d e^{3}{\rm sign}\left (b x + a\right ) + 60 \, A a^{3} e^{4}{\rm sign}\left (b x + a\right )\right )} e^{\left (-5\right )}}{420 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(B*x + A)/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/420*(140*B*b^3*x^4*e^4*sign(b*x + a) + 140*B*b^3*d*x^3*e^3*sign(b*x + a) + 84
*B*b^3*d^2*x^2*e^2*sign(b*x + a) + 28*B*b^3*d^3*x*e*sign(b*x + a) + 4*B*b^3*d^4*
sign(b*x + a) + 315*B*a*b^2*x^3*e^4*sign(b*x + a) + 105*A*b^3*x^3*e^4*sign(b*x +
a) + 189*B*a*b^2*d*x^2*e^3*sign(b*x + a) + 63*A*b^3*d*x^2*e^3*sign(b*x + a) + 6
3*B*a*b^2*d^2*x*e^2*sign(b*x + a) + 21*A*b^3*d^2*x*e^2*sign(b*x + a) + 9*B*a*b^2
*d^3*e*sign(b*x + a) + 3*A*b^3*d^3*e*sign(b*x + a) + 252*B*a^2*b*x^2*e^4*sign(b*
x + a) + 252*A*a*b^2*x^2*e^4*sign(b*x + a) + 84*B*a^2*b*d*x*e^3*sign(b*x + a) +
84*A*a*b^2*d*x*e^3*sign(b*x + a) + 12*B*a^2*b*d^2*e^2*sign(b*x + a) + 12*A*a*b^2
*d^2*e^2*sign(b*x + a) + 70*B*a^3*x*e^4*sign(b*x + a) + 210*A*a^2*b*x*e^4*sign(b
*x + a) + 10*B*a^3*d*e^3*sign(b*x + a) + 30*A*a^2*b*d*e^3*sign(b*x + a) + 60*A*a
^3*e^4*sign(b*x + a))*e^(-5)/(x*e + d)^7