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

Optimal. Leaf size=210 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 x^6 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{4 x^4 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.244145, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+3 A b)}{6 x^6 (a+b x)}-\frac{3 a b \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{5 x^5 (a+b x)}-\frac{b^2 \sqrt{a^2+2 a b x+b^2 x^2} (3 a B+A b)}{4 x^4 (a+b x)}-\frac{b^3 B \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)}-\frac{a^3 A \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.0673, size = 173, normalized size = 0.82 \[ - \frac{A \left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{14 a x^{7}} + \frac{\left (2 a + 2 b x\right ) \left (3 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{84 a^{2} x^{6}} - \frac{b \left (2 a + 2 b x\right ) \left (3 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}}{168 a^{3} x^{5}} + \frac{b \left (3 A b - 7 B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{420 a^{4} x^{5}} \]

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)/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.0549577, size = 87, normalized size = 0.41 \[ -\frac{\sqrt{(a+b x)^2} \left (10 a^3 (6 A+7 B x)+42 a^2 b x (5 A+6 B x)+63 a b^2 x^2 (4 A+5 B x)+35 b^3 x^3 (3 A+4 B x)\right )}{420 x^7 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[(a + b*x)^2]*(35*b^3*x^3*(3*A + 4*B*x) + 63*a*b^2*x^2*(4*A + 5*B*x) + 42*
a^2*b*x*(5*A + 6*B*x) + 10*a^3*(6*A + 7*B*x)))/(420*x^7*(a + b*x))

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Maple [A]  time = 0.01, size = 92, normalized size = 0.4 \[ -{\frac{140\,B{x}^{4}{b}^{3}+105\,A{b}^{3}{x}^{3}+315\,B{x}^{3}a{b}^{2}+252\,A{x}^{2}a{b}^{2}+252\,B{x}^{2}{a}^{2}b+210\,A{a}^{2}bx+70\,{a}^{3}Bx+60\,A{a}^{3}}{420\,{x}^{7} \left ( bx+a \right ) ^{3}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \]

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)/x^8,x)

[Out]

-1/420*(140*B*b^3*x^4+105*A*b^3*x^3+315*B*a*b^2*x^3+252*A*a*b^2*x^2+252*B*a^2*b*
x^2+210*A*a^2*b*x+70*B*a^3*x+60*A*a^3)*((b*x+a)^2)^(3/2)/x^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)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.275218, size = 99, normalized size = 0.47 \[ -\frac{140 \, B b^{3} x^{4} + 60 \, A a^{3} + 105 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 252 \,{\left (B a^{2} b + A a b^{2}\right )} x^{2} + 70 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{420 \, x^{7}} \]

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)/x^8,x, algorithm="fricas")

[Out]

-1/420*(140*B*b^3*x^4 + 60*A*a^3 + 105*(3*B*a*b^2 + A*b^3)*x^3 + 252*(B*a^2*b +
A*a*b^2)*x^2 + 70*(B*a^3 + 3*A*a^2*b)*x)/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{8}}\, dx \]

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)/x**8,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(3/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.2742, size = 201, normalized size = 0.96 \[ -\frac{{\left (7 \, B a b^{6} - 3 \, A b^{7}\right )}{\rm sign}\left (b x + a\right )}{420 \, a^{4}} - \frac{140 \, B b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 315 \, B a b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 105 \, A b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 252 \, B a^{2} b x^{2}{\rm sign}\left (b x + a\right ) + 252 \, A a b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 70 \, B a^{3} x{\rm sign}\left (b x + a\right ) + 210 \, A a^{2} b x{\rm sign}\left (b x + a\right ) + 60 \, A a^{3}{\rm sign}\left (b x + a\right )}{420 \, x^{7}} \]

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)/x^8,x, algorithm="giac")

[Out]

-1/420*(7*B*a*b^6 - 3*A*b^7)*sign(b*x + a)/a^4 - 1/420*(140*B*b^3*x^4*sign(b*x +
a) + 315*B*a*b^2*x^3*sign(b*x + a) + 105*A*b^3*x^3*sign(b*x + a) + 252*B*a^2*b*
x^2*sign(b*x + a) + 252*A*a*b^2*x^2*sign(b*x + a) + 70*B*a^3*x*sign(b*x + a) + 2
10*A*a^2*b*x*sign(b*x + a) + 60*A*a^3*sign(b*x + a))/x^7