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

Optimal. Leaf size=83 \[ \frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0240749, antiderivative size = 83, 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} \[ \frac{a^2 x^2}{b^3 \sqrt{c x^2}}-\frac{a^3 x \log (a+b x)}{b^4 \sqrt{c x^2}}-\frac{a x^3}{2 b^2 \sqrt{c x^2}}+\frac{x^4}{3 b \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[c*x^2]*(a + b*x)),x]

[Out]

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

Mathematica [A]  time = 0.0103318, size = 51, normalized size = 0.61 \[ \frac{x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 50, normalized size = 0.6 \begin{align*} -{\frac{x \left ( -2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx \right ) }{6\,{b}^{4}}{\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)/(c*x^2)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.82468, size = 116, normalized size = 1.4 \begin{align*} \frac{{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^3*x^3 - 3*a*b^2*x^2 + 6*a^2*b*x - 6*a^3*log(b*x + a))*sqrt(c*x^2)/(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 )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.10552, size = 108, normalized size = 1.3 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2}}{\left (x{\left (\frac{2 \, x}{b c} - \frac{3 \, a}{b^{2} c}\right )} + \frac{6 \, a^{2}}{b^{3} c}\right )} + \frac{a^{3} \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{b^{4} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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