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

Optimal. Leaf size=64 \[ -\frac{a^2 x}{b^3 \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 \sqrt{c x^2}}+\frac{x^2}{b^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0196238, antiderivative size = 64, 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} \[ -\frac{a^2 x}{b^3 \sqrt{c x^2} (a+b x)}-\frac{2 a x \log (a+b x)}{b^3 \sqrt{c x^2}}+\frac{x^2}{b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

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

Mathematica [A]  time = 0.0119881, size = 52, normalized size = 0.81 \[ \frac{x \left (-a^2+a b x-2 a (a+b x) \log (a+b x)+b^2 x^2\right )}{b^3 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 60, normalized size = 0.9 \begin{align*} -{\frac{x \left ( 2\,\ln \left ( bx+a \right ) xab-{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -abx+{a}^{2} \right ) }{{b}^{3} \left ( bx+a \right ) }{\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)^2/(c*x^2)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.34299, size = 124, normalized size = 1.94 \begin{align*} \frac{{\left (b^{2} x^{2} + a b x - a^{2} - 2 \,{\left (a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{b^{4} c x^{2} + a b^{3} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x^2 + a*b*x - a^2 - 2*(a*b*x + a^2)*log(b*x + a))*sqrt(c*x^2)/(b^4*c*x^2 + a*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 )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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