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

Optimal. Leaf size=35 \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.005395, antiderivative size = 35, 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} \[ \frac{a x \log (x)}{c \sqrt{c x^2}}+\frac{b x^2}{c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(a + b*x))/(c*x^2)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0039511, size = 21, normalized size = 0.6 \[ \frac{x^3 (a \log (x)+b x)}{\left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(b*x + a*Log[x]))/(c*x^2)^(3/2)

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Maple [A]  time = 0.002, size = 20, normalized size = 0.6 \begin{align*}{{x}^{3} \left ( bx+a\ln \left ( x \right ) \right ) \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07308, size = 31, normalized size = 0.89 \begin{align*} \frac{b x^{2}}{\sqrt{c x^{2}} c} + \frac{a \log \left (x\right )}{c^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60319, size = 51, normalized size = 1.46 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x + a \log \left (x\right )\right )}}{c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c^2*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06013, size = 54, normalized size = 1.54 \begin{align*} -\frac{\frac{a \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \frac{\sqrt{c x^{2}} b}{c}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*log(abs(-sqrt(c)*x + sqrt(c*x^2)))/sqrt(c) - sqrt(c*x^2)*b/c)/c