3.864 \(\int \frac{(c x^2)^{3/2}}{x^3 (a+b x)} \, dx\)

Optimal. Leaf size=23 \[ \frac{c \sqrt{c x^2} \log (a+b x)}{b x} \]

[Out]

(c*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

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Rubi [A]  time = 0.0038999, antiderivative size = 23, 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, Rules used = {15, 31} \[ \frac{c \sqrt{c x^2} \log (a+b x)}{b x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(x^3*(a + b*x)),x]

[Out]

(c*Sqrt[c*x^2]*Log[a + b*x])/(b*x)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\left (c x^2\right )^{3/2}}{x^3 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{a+b x} \, dx}{x}\\ &=\frac{c \sqrt{c x^2} \log (a+b x)}{b x}\\ \end{align*}

Mathematica [A]  time = 0.0032357, size = 22, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} \log (a+b x)}{b x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(3/2)/(x^3*(a + b*x)),x]

[Out]

((c*x^2)^(3/2)*Log[a + b*x])/(b*x^3)

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Maple [A]  time = 0.002, size = 21, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( bx+a \right ) }{b{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/x^3/(b*x+a),x)

[Out]

(c*x^2)^(3/2)/x^3*ln(b*x+a)/b

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Maxima [A]  time = 1.05648, size = 18, normalized size = 0.78 \begin{align*} \frac{c^{\frac{3}{2}} \log \left (b x + a\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^3/(b*x+a),x, algorithm="maxima")

[Out]

c^(3/2)*log(b*x + a)/b

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Fricas [A]  time = 1.57137, size = 46, normalized size = 2. \begin{align*} \frac{\sqrt{c x^{2}} c \log \left (b x + a\right )}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^3/(b*x+a),x, algorithm="fricas")

[Out]

sqrt(c*x^2)*c*log(b*x + a)/(b*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{3} \left (a + b x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)/x**3/(b*x+a),x)

[Out]

Integral((c*x**2)**(3/2)/(x**3*(a + b*x)), x)

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Giac [A]  time = 1.05631, size = 38, normalized size = 1.65 \begin{align*} c^{\frac{3}{2}}{\left (\frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b} - \frac{\log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^3/(b*x+a),x, algorithm="giac")

[Out]

c^(3/2)*(log(abs(b*x + a))*sgn(x)/b - log(abs(a))*sgn(x)/b)