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

Optimal. Leaf size=48 \[ \frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8} \]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(9*a*x^8) + (4*b*(a*x + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi [A]  time = 0.0610288, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{4 b \left (a x+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{2 \left (a x+b x^2\right )^{7/2}}{9 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a*x + b*x^2)^(5/2)/x^8,x]

[Out]

(-2*(a*x + b*x^2)^(7/2))/(9*a*x^8) + (4*b*(a*x + b*x^2)^(7/2))/(63*a^2*x^7)

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Rubi in Sympy [A]  time = 6.07118, size = 42, normalized size = 0.88 \[ - \frac{2 \left (a x + b x^{2}\right )^{\frac{7}{2}}}{9 a x^{8}} + \frac{4 b \left (a x + b x^{2}\right )^{\frac{7}{2}}}{63 a^{2} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a*x)**(5/2)/x**8,x)

[Out]

-2*(a*x + b*x**2)**(7/2)/(9*a*x**8) + 4*b*(a*x + b*x**2)**(7/2)/(63*a**2*x**7)

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Mathematica [A]  time = 0.0337755, size = 36, normalized size = 0.75 \[ \frac{2 (a+b x)^3 \sqrt{x (a+b x)} (2 b x-7 a)}{63 a^2 x^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^3*Sqrt[x*(a + b*x)]*(-7*a + 2*b*x))/(63*a^2*x^5)

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Maple [A]  time = 0.006, size = 33, normalized size = 0.7 \[ -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -2\,bx+7\,a \right ) }{63\,{x}^{7}{a}^{2}} \left ( b{x}^{2}+ax \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a*x)^(5/2)/x^8,x)

[Out]

-2/63*(b*x+a)*(-2*b*x+7*a)*(b*x^2+a*x)^(5/2)/x^7/a^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229657, size = 81, normalized size = 1.69 \[ \frac{2 \,{\left (2 \, b^{4} x^{4} - a b^{3} x^{3} - 15 \, a^{2} b^{2} x^{2} - 19 \, a^{3} b x - 7 \, a^{4}\right )} \sqrt{b x^{2} + a x}}{63 \, a^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(2*b^4*x^4 - a*b^3*x^3 - 15*a^2*b^2*x^2 - 19*a^3*b*x - 7*a^4)*sqrt(b*x^2 +
a*x)/(a^2*x^5)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a*x)**(5/2)/x**8,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216819, size = 301, normalized size = 6.27 \[ \frac{2 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{7} b^{\frac{7}{2}} + 273 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{6} a b^{3} + 567 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{5} a^{2} b^{\frac{5}{2}} + 693 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{4} a^{3} b^{2} + 525 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{3} a^{4} b^{\frac{3}{2}} + 243 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{2} a^{5} b + 63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )} a^{6} \sqrt{b} + 7 \, a^{7}\right )}}{63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a x}\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/63*(63*(sqrt(b)*x - sqrt(b*x^2 + a*x))^7*b^(7/2) + 273*(sqrt(b)*x - sqrt(b*x^2
+ a*x))^6*a*b^3 + 567*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^2*b^(5/2) + 693*(sqrt
(b)*x - sqrt(b*x^2 + a*x))^4*a^3*b^2 + 525*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^4
*b^(3/2) + 243*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^5*b + 63*(sqrt(b)*x - sqrt(b*
x^2 + a*x))*a^6*sqrt(b) + 7*a^7)/(sqrt(b)*x - sqrt(b*x^2 + a*x))^9