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

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

[Out]

(x*Log[a + b*x])/(b*c*Sqrt[c*x^2])

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Rubi [A]  time = 0.0039192, 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} \[ \frac{x \log (a+b x)}{b c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

(x*Log[a + b*x])/(b*c*Sqrt[c*x^2])

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

Mathematica [A]  time = 0.0037252, size = 22, normalized size = 0.96 \[ \frac{x^3 \log (a+b x)}{b \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*Log[a + b*x])/(b*(c*x^2)^(3/2))

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Maple [A]  time = 0.001, size = 21, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}\ln \left ( bx+a \right ) }{b} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44449, size = 49, normalized size = 2.13 \begin{align*} \frac{\sqrt{c x^{2}} \log \left (b x + a\right )}{b c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*log(b*x + a)/(b*c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07928, size = 47, normalized size = 2.04 \begin{align*} -\frac{\log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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