3.697 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^8} \, dx\)

Optimal. Leaf size=77 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{6 a^2 x^6}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7} \]

[Out]

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

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Rubi [A]  time = 0.156003, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2} (A b-a B)}{6 a^2 x^6}-\frac{A \left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 a^2 x^7} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 23.9879, size = 73, normalized size = 0.95 \[ - \frac{A \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{7}{2}}}{7 a^{2} x^{7}} + \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}}{12 a^{2} x^{6}} \]

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)**(5/2)/x**8,x)

[Out]

-A*(a**2 + 2*a*b*x + b**2*x**2)**(7/2)/(7*a**2*x**7) + (2*a + 2*b*x)*(A*b - B*a)
*(a**2 + 2*a*b*x + b**2*x**2)**(5/2)/(12*a**2*x**6)

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Mathematica [A]  time = 0.079286, size = 122, normalized size = 1.58 \[ -\frac{\sqrt{(a+b x)^2} \left (a^5 (6 A+7 B x)+7 a^4 b x (5 A+6 B x)+21 a^3 b^2 x^2 (4 A+5 B x)+35 a^2 b^3 x^3 (3 A+4 B x)+35 a b^4 x^4 (2 A+3 B x)+21 b^5 x^5 (A+2 B x)\right )}{42 x^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/x^8,x]

[Out]

-(Sqrt[(a + b*x)^2]*(21*b^5*x^5*(A + 2*B*x) + 35*a*b^4*x^4*(2*A + 3*B*x) + 35*a^
2*b^3*x^3*(3*A + 4*B*x) + 21*a^3*b^2*x^2*(4*A + 5*B*x) + 7*a^4*b*x*(5*A + 6*B*x)
+ a^5*(6*A + 7*B*x)))/(42*x^7*(a + b*x))

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Maple [B]  time = 0.011, size = 140, normalized size = 1.8 \[ -{\frac{42\,B{b}^{5}{x}^{6}+21\,A{x}^{5}{b}^{5}+105\,B{x}^{5}a{b}^{4}+70\,A{x}^{4}a{b}^{4}+140\,B{x}^{4}{a}^{2}{b}^{3}+105\,A{x}^{3}{a}^{2}{b}^{3}+105\,B{x}^{3}{a}^{3}{b}^{2}+84\,A{x}^{2}{a}^{3}{b}^{2}+42\,B{x}^{2}{a}^{4}b+35\,Ax{a}^{4}b+7\,Bx{a}^{5}+6\,A{a}^{5}}{42\,{x}^{7} \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2)/x^8,x)

[Out]

-1/42*(42*B*b^5*x^6+21*A*b^5*x^5+105*B*a*b^4*x^5+70*A*a*b^4*x^4+140*B*a^2*b^3*x^
4+105*A*a^2*b^3*x^3+105*B*a^3*b^2*x^3+84*A*a^3*b^2*x^2+42*B*a^4*b*x^2+35*A*a^4*b
*x+7*B*a^5*x+6*A*a^5)*((b*x+a)^2)^(5/2)/x^7/(b*x+a)^5

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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)^(5/2)*(B*x + A)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274647, size = 161, normalized size = 2.09 \[ -\frac{42 \, B b^{5} x^{6} + 6 \, A a^{5} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 70 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 105 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 42 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^8,x, algorithm="fricas")

[Out]

-1/42*(42*B*b^5*x^6 + 6*A*a^5 + 21*(5*B*a*b^4 + A*b^5)*x^5 + 70*(2*B*a^2*b^3 + A
*a*b^4)*x^4 + 105*(B*a^3*b^2 + A*a^2*b^3)*x^3 + 42*(B*a^4*b + 2*A*a^3*b^2)*x^2 +
7*(B*a^5 + 5*A*a^4*b)*x)/x^7

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}{x^{8}}\, dx \]

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)**(5/2)/x**8,x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**8, x)

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GIAC/XCAS [A]  time = 0.273913, size = 298, normalized size = 3.87 \[ -\frac{{\left (7 \, B a b^{6} - A b^{7}\right )}{\rm sign}\left (b x + a\right )}{42 \, a^{2}} - \frac{42 \, B b^{5} x^{6}{\rm sign}\left (b x + a\right ) + 105 \, B a b^{4} x^{5}{\rm sign}\left (b x + a\right ) + 21 \, A b^{5} x^{5}{\rm sign}\left (b x + a\right ) + 140 \, B a^{2} b^{3} x^{4}{\rm sign}\left (b x + a\right ) + 70 \, A a b^{4} x^{4}{\rm sign}\left (b x + a\right ) + 105 \, B a^{3} b^{2} x^{3}{\rm sign}\left (b x + a\right ) + 105 \, A a^{2} b^{3} x^{3}{\rm sign}\left (b x + a\right ) + 42 \, B a^{4} b x^{2}{\rm sign}\left (b x + a\right ) + 84 \, A a^{3} b^{2} x^{2}{\rm sign}\left (b x + a\right ) + 7 \, B a^{5} x{\rm sign}\left (b x + a\right ) + 35 \, A a^{4} b x{\rm sign}\left (b x + a\right ) + 6 \, A a^{5}{\rm sign}\left (b x + a\right )}{42 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(B*x + A)/x^8,x, algorithm="giac")

[Out]

-1/42*(7*B*a*b^6 - A*b^7)*sign(b*x + a)/a^2 - 1/42*(42*B*b^5*x^6*sign(b*x + a) +
105*B*a*b^4*x^5*sign(b*x + a) + 21*A*b^5*x^5*sign(b*x + a) + 140*B*a^2*b^3*x^4*
sign(b*x + a) + 70*A*a*b^4*x^4*sign(b*x + a) + 105*B*a^3*b^2*x^3*sign(b*x + a) +
105*A*a^2*b^3*x^3*sign(b*x + a) + 42*B*a^4*b*x^2*sign(b*x + a) + 84*A*a^3*b^2*x
^2*sign(b*x + a) + 7*B*a^5*x*sign(b*x + a) + 35*A*a^4*b*x*sign(b*x + a) + 6*A*a^
5*sign(b*x + a))/x^7