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

Optimal. Leaf size=61 \[ \frac{a^2 x \log (a+b x)}{b^3 \sqrt{c x^2}}-\frac{a x^2}{b^2 \sqrt{c x^2}}+\frac{x^3}{2 b \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0170711, antiderivative size = 61, 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, Rules used = {15, 43} \[ \frac{a^2 x \log (a+b x)}{b^3 \sqrt{c x^2}}-\frac{a x^2}{b^2 \sqrt{c x^2}}+\frac{x^3}{2 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

-((a*x^2)/(b^2*Sqrt[c*x^2])) + x^3/(2*b*Sqrt[c*x^2]) + (a^2*x*Log[a + b*x])/(b^3*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0122701, size = 39, normalized size = 0.64 \[ \frac{x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 38, normalized size = 0.6 \begin{align*}{\frac{x \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx \right ) }{2\,{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/(b*x+a)/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.80387, size = 92, normalized size = 1.51 \begin{align*} \frac{{\left (b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(b^2*x^2 - 2*a*b*x + 2*a^2*log(b*x + a))*sqrt(c*x^2)/(b^3*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.06914, size = 89, normalized size = 1.46 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2}}{\left (\frac{x}{b c} - \frac{2 \, a}{b^{2} c}\right )} - \frac{a^{2} \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{3} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(b*x+a)/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(c*x^2)*(x/(b*c) - 2*a/(b^2*c)) - a^2*log(abs(-(sqrt(c)*x - sqrt(c*x^2))*b - 2*a*sqrt(c)))/(b^3*sqrt(c
))