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

Optimal. Leaf size=107 \[ -\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b} \]

[Out]

-((a^3*c*Sqrt[c*x^2])/b^4) + (a^2*c*x*Sqrt[c*x^2])/(2*b^3) - (a*c*x^2*Sqrt[c*x^2])/(3*b^2) + (c*x^3*Sqrt[c*x^2
])/(4*b) + (a^4*c*Sqrt[c*x^2]*Log[a + b*x])/(b^5*x)

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Rubi [A]  time = 0.0321806, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ -\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c*x^2)^(3/2))/(a + b*x),x]

[Out]

-((a^3*c*Sqrt[c*x^2])/b^4) + (a^2*c*x*Sqrt[c*x^2])/(2*b^3) - (a*c*x^2*Sqrt[c*x^2])/(3*b^2) + (c*x^3*Sqrt[c*x^2
])/(4*b) + (a^4*c*Sqrt[c*x^2]*Log[a + b*x])/(b^5*x)

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 \left (c x^2\right )^{3/2}}{a+b x} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^4}{a+b x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{a^3 c \sqrt{c x^2}}{b^4}+\frac{a^2 c x \sqrt{c x^2}}{2 b^3}-\frac{a c x^2 \sqrt{c x^2}}{3 b^2}+\frac{c x^3 \sqrt{c x^2}}{4 b}+\frac{a^4 c \sqrt{c x^2} \log (a+b x)}{b^5 x}\\ \end{align*}

Mathematica [A]  time = 0.013049, size = 64, normalized size = 0.6 \[ \frac{\left (c x^2\right )^{3/2} \left (b x \left (6 a^2 b x-12 a^3-4 a b^2 x^2+3 b^3 x^3\right )+12 a^4 \log (a+b x)\right )}{12 b^5 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*x^2)^(3/2)*(b*x*(-12*a^3 + 6*a^2*b*x - 4*a*b^2*x^2 + 3*b^3*x^3) + 12*a^4*Log[a + b*x]))/(12*b^5*x^3)

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Maple [A]  time = 0.004, size = 63, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{4}{x}^{4}-4\,{x}^{3}a{b}^{3}+6\,{x}^{2}{a}^{2}{b}^{2}+12\,{a}^{4}\ln \left ( bx+a \right ) -12\,bx{a}^{3}}{12\,{b}^{5}{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(c*x^2)^(3/2)*(3*b^4*x^4-4*x^3*a*b^3+6*x^2*a^2*b^2+12*a^4*ln(b*x+a)-12*b*x*a^3)/x^3/b^5

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.60599, size = 153, normalized size = 1.43 \begin{align*} \frac{{\left (3 \, b^{4} c x^{4} - 4 \, a b^{3} c x^{3} + 6 \, a^{2} b^{2} c x^{2} - 12 \, a^{3} b c x + 12 \, a^{4} c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{12 \, b^{5} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*c*x^4 - 4*a*b^3*c*x^3 + 6*a^2*b^2*c*x^2 - 12*a^3*b*c*x + 12*a^4*c*log(b*x + a))*sqrt(c*x^2)/(b^5*x
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05754, size = 109, normalized size = 1.02 \begin{align*} \frac{1}{12} \, c^{\frac{3}{2}}{\left (\frac{12 \, a^{4} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} - \frac{12 \, a^{4} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} \mathrm{sgn}\left (x\right ) - 4 \, a b^{2} x^{3} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} b x^{2} \mathrm{sgn}\left (x\right ) - 12 \, a^{3} x \mathrm{sgn}\left (x\right )}{b^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*c^(3/2)*(12*a^4*log(abs(b*x + a))*sgn(x)/b^5 - 12*a^4*log(abs(a))*sgn(x)/b^5 + (3*b^3*x^4*sgn(x) - 4*a*b^
2*x^3*sgn(x) + 6*a^2*b*x^2*sgn(x) - 12*a^3*x*sgn(x))/b^4)