3.876 \(\int \frac{\left (c x^2\right )^{5/2}}{x^7 (a+b x)} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b c^2 \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c^2 \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c^2 \sqrt{c x^2}}{a x^2} \]

[Out]

-((c^2*Sqrt[c*x^2])/(a*x^2)) - (b*c^2*Sqrt[c*x^2]*Log[x])/(a^2*x) + (b*c^2*Sqrt[
c*x^2]*Log[a + b*x])/(a^2*x)

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Rubi [A]  time = 0.0474253, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{b c^2 \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b c^2 \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{c^2 \sqrt{c x^2}}{a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(5/2)/(x^7*(a + b*x)),x]

[Out]

-((c^2*Sqrt[c*x^2])/(a*x^2)) - (b*c^2*Sqrt[c*x^2]*Log[x])/(a^2*x) + (b*c^2*Sqrt[
c*x^2]*Log[a + b*x])/(a^2*x)

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Rubi in Sympy [A]  time = 19.491, size = 63, normalized size = 0.9 \[ - \frac{c^{2} \sqrt{c x^{2}}}{a x^{2}} - \frac{b c^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{2} x} + \frac{b c^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(5/2)/x**7/(b*x+a),x)

[Out]

-c**2*sqrt(c*x**2)/(a*x**2) - b*c**2*sqrt(c*x**2)*log(x)/(a**2*x) + b*c**2*sqrt(
c*x**2)*log(a + b*x)/(a**2*x)

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Mathematica [A]  time = 0.0198117, size = 34, normalized size = 0.49 \[ -\frac{c^3 (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*x^2)^(5/2)/(x^7*(a + b*x)),x]

[Out]

-((c^3*(a + b*x*Log[x] - b*x*Log[a + b*x]))/(a^2*Sqrt[c*x^2]))

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Maple [A]  time = 0.007, size = 33, normalized size = 0.5 \[ -{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a}{{x}^{6}{a}^{2}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)/x^7/(b*x+a),x)

[Out]

-(c*x^2)^(5/2)*(b*ln(x)*x-b*ln(b*x+a)*x+a)/x^6/a^2

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Maxima [A]  time = 1.32117, size = 50, normalized size = 0.71 \[ \frac{b c^{\frac{5}{2}} \log \left (b x + a\right )}{a^{2}} - \frac{b c^{\frac{5}{2}} \log \left (x\right )}{a^{2}} - \frac{c^{\frac{5}{2}}}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)/((b*x + a)*x^7),x, algorithm="maxima")

[Out]

b*c^(5/2)*log(b*x + a)/a^2 - b*c^(5/2)*log(x)/a^2 - c^(5/2)/(a*x)

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Fricas [A]  time = 0.215148, size = 50, normalized size = 0.71 \[ \frac{{\left (b c^{2} x \log \left (\frac{b x + a}{x}\right ) - a c^{2}\right )} \sqrt{c x^{2}}}{a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)/((b*x + a)*x^7),x, algorithm="fricas")

[Out]

(b*c^2*x*log((b*x + a)/x) - a*c^2)*sqrt(c*x^2)/(a^2*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}}{x^{7} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(5/2)/x**7/(b*x+a),x)

[Out]

Integral((c*x**2)**(5/2)/(x**7*(a + b*x)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)/((b*x + a)*x^7),x, algorithm="giac")

[Out]

Exception raised: TypeError