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

Optimal. Leaf size=84 \[ \frac{a^2 c \sqrt{c x^2}}{b^3}-\frac{a^3 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{a c x \sqrt{c x^2}}{2 b^2}+\frac{c x^2 \sqrt{c x^2}}{3 b} \]

[Out]

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

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Rubi [A]  time = 0.0246509, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac{a^2 c \sqrt{c x^2}}{b^3}-\frac{a^3 c \sqrt{c x^2} \log (a+b x)}{b^4 x}-\frac{a c x \sqrt{c x^2}}{2 b^2}+\frac{c x^2 \sqrt{c x^2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(a + b*x),x]

[Out]

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

Mathematica [A]  time = 0.0114378, size = 53, normalized size = 0.63 \[ \frac{\left (c x^2\right )^{3/2} \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(3/2)/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.005, size = 52, normalized size = 0.6 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx}{6\,{x}^{3}{b}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59563, size = 124, normalized size = 1.48 \begin{align*} \frac{{\left (2 \, b^{3} c x^{3} - 3 \, a b^{2} c x^{2} + 6 \, a^{2} b c x - 6 \, a^{3} c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.05621, size = 93, normalized size = 1.11 \begin{align*} -\frac{1}{6} \, c^{\frac{3}{2}}{\left (\frac{6 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} - \frac{6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{4}} - \frac{2 \, b^{2} x^{3} \mathrm{sgn}\left (x\right ) - 3 \, a b x^{2} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} x \mathrm{sgn}\left (x\right )}{b^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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