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

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

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Rubi [A]  time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \begin {gather*} \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}} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 0.56 \begin {gather*} \frac {x^3 \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \left (c x^2\right )^{3/2}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.03, size = 39, normalized size = 0.64 \begin {gather*} \frac {a^2 x^3 \log (x)+\frac {1}{2} \left (4 a b x^4+b^2 x^5\right )}{\left (c x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.18, size = 35, normalized size = 0.57 \begin {gather*} \frac {{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x)\right )} \sqrt {c x^{2}}}{2 \, c^{2} x} \end {gather*}

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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giac [A]  time = 1.10, size = 55, normalized size = 0.90 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 33, normalized size = 0.54 \begin {gather*} \frac {\left (b^{2} x^{2}+2 a^{2} \ln \relax (x )+4 a b x \right ) x^{3}}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}

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 = 1.33, size = 45, normalized size = 0.74 \begin {gather*} \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 \relax (x)}{c^{\frac {3}{2}}} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^2\,{\left (a+b\,x\right )}^2}{{\left (c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b x\right )^{2}}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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