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

Optimal. Leaf size=25 \[ \frac{c^2 \sqrt{c x^2} \log (a+b x)}{b x} \]

[Out]

(c^2*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi [A]  time = 0.0123549, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{c^2 \sqrt{c x^2} \log (a+b x)}{b x} \]

Antiderivative was successfully verified.

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

[Out]

(c^2*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi in Sympy [A]  time = 12.351, size = 20, normalized size = 0.8 \[ \frac{c^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c**2*sqrt(c*x**2)*log(a + b*x)/(b*x)

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Mathematica [A]  time = 0.0063219, size = 22, normalized size = 0.88 \[ \frac{\left (c x^2\right )^{5/2} \log (a+b x)}{b x^5} \]

Antiderivative was successfully verified.

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

[Out]

((c*x^2)^(5/2)*Log[a + b*x])/(b*x^5)

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Maple [A]  time = 0.004, size = 21, normalized size = 0.8 \[{\frac{\ln \left ( bx+a \right ) }{b{x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3536, size = 18, normalized size = 0.72 \[ \frac{c^{\frac{5}{2}} \log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(5/2)*log(b*x + a)/b

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Fricas [A]  time = 0.214451, size = 31, normalized size = 1.24 \[ \frac{\sqrt{c x^{2}} c^{2} \log \left (b x + a\right )}{b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^2)*c^2*log(b*x + a)/(b*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.20629, size = 46, normalized size = 1.84 \[{\left (\frac{c^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b} - \frac{c^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b}\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c^2*ln(abs(b*x + a))*sign(x)/b - c^2*ln(abs(a))*sign(x)/b)*sqrt(c)