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

Optimal. Leaf size=61 \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0109616, antiderivative size = 61, 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} \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0072139, size = 34, normalized size = 0.56 \[ \frac{x^3 \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 33, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx \right ) }{2} \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)^2/(c*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.04843, size = 61, normalized size = 1. \begin{align*} \frac{b^{2} x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{2 \, a b x^{2}}{\sqrt{c x^{2}} c} + \frac{a^{2} \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)^2/(c*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.54526, size = 81, normalized size = 1.33 \begin{align*} \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^2*x^2 + 4*a*b*x + 2*a^2*log(x))*sqrt(c*x^2)/(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 )^{2}}{\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)**2/(c*x**2)**(3/2),x)

[Out]

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

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Giac [A]  time = 1.07647, size = 74, normalized size = 1.21 \begin{align*} -\frac{\frac{2 \, a^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \sqrt{c x^{2}}{\left (\frac{b^{2} x}{c} + \frac{4 \, a b}{c}\right )}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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