3.910 \(\int \frac{x^4}{\sqrt{c x^2} (a+b x)^2} \, dx\)

Optimal. Leaf size=86 \[ \frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}} \]

[Out]

(-2*a*x^2)/(b^3*Sqrt[c*x^2]) + x^3/(2*b^2*Sqrt[c*x^2]) + (a^3*x)/(b^4*Sqrt[c*x^2]*(a + b*x)) + (3*a^2*x*Log[a
+ b*x])/(b^4*Sqrt[c*x^2])

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Rubi [A]  time = 0.02566, antiderivative size = 86, 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} \[ \frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

(-2*a*x^2)/(b^3*Sqrt[c*x^2]) + x^3/(2*b^2*Sqrt[c*x^2]) + (a^3*x)/(b^4*Sqrt[c*x^2]*(a + b*x)) + (3*a^2*x*Log[a
+ b*x])/(b^4*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^4}{\sqrt{c x^2} (a+b x)^2} \, dx &=\frac{x \int \frac{x^3}{(a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{2 a x^2}{b^3 \sqrt{c x^2}}+\frac{x^3}{2 b^2 \sqrt{c x^2}}+\frac{a^3 x}{b^4 \sqrt{c x^2} (a+b x)}+\frac{3 a^2 x \log (a+b x)}{b^4 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0129685, size = 69, normalized size = 0.8 \[ \frac{x \left (-4 a^2 b x+6 a^2 (a+b x) \log (a+b x)+2 a^3-3 a b^2 x^2+b^3 x^3\right )}{2 b^4 \sqrt{c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(Sqrt[c*x^2]*(a + b*x)^2),x]

[Out]

(x*(2*a^3 - 4*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 + 6*a^2*(a + b*x)*Log[a + b*x]))/(2*b^4*Sqrt[c*x^2]*(a + b*x))

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Maple [A]  time = 0.004, size = 74, normalized size = 0.9 \begin{align*}{\frac{x \left ({b}^{3}{x}^{3}+6\,\ln \left ( bx+a \right ) x{a}^{2}b-3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -4\,{a}^{2}bx+2\,{a}^{3} \right ) }{2\,{b}^{4} \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^4/(b*x+a)^2/(c*x^2)^(1/2),x)

[Out]

1/2*x*(b^3*x^3+6*ln(b*x+a)*x*a^2*b-3*a*b^2*x^2+6*a^3*ln(b*x+a)-4*a^2*b*x+2*a^3)/(c*x^2)^(1/2)/b^4/(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^4/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.16849, size = 159, normalized size = 1.85 \begin{align*} \frac{{\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 2 \, a^{3} + 6 \,{\left (a^{2} b x + a^{3}\right )} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (b^{5} c x^{2} + a b^{4} c x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

1/2*(b^3*x^3 - 3*a*b^2*x^2 - 4*a^2*b*x + 2*a^3 + 6*(a^2*b*x + a^3)*log(b*x + a))*sqrt(c*x^2)/(b^5*c*x^2 + a*b^
4*c*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\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**4/(b*x+a)**2/(c*x**2)**(1/2),x)

[Out]

Integral(x**4/(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^{4}}{\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^4/(b*x+a)^2/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

integrate(x^4/(sqrt(c*x^2)*(b*x + a)^2), x)