3.9.37 \(\int \frac {x^2 (a+b x)^2}{(c x^2)^{3/2}} \, dx\) [837]

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

[Out]

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

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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, 45} \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 45

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 (b x (4 a+b x)+2 a^2 \log (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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Maple [A]
time = 0.12, size = 33, normalized size = 0.54

method	result	size
default	\(\frac {x^{3} \left (x^{2} b^{2}+2 a^{2} \ln \left (x \right )+4 a b x \right )}{2 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(33\)
risch	\(\frac {x b \left (\frac {1}{2} x^{2} b +2 a x \right )}{c \sqrt {c \,x^{2}}}+\frac {a^{2} x \ln \left (x \right )}{c \sqrt {c \,x^{2}}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, 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 \left (x\right )}{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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Fricas [A]
time = 0.47, size = 35, normalized size = 0.57 \begin {gather*} \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 {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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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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Giac [A]
time = 0.84, size = 48, normalized size = 0.79 \begin {gather*} \frac {\frac {2 \, a^{2} \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {b^{2} c^{\frac {3}{2}} x^{2} \mathrm {sgn}\left (x\right ) + 4 \, a b c^{\frac {3}{2}} x \mathrm {sgn}\left (x\right )}{c^{2}}}{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(x))/(sqrt(c)*sgn(x)) + (b^2*c^(3/2)*x^2*sgn(x) + 4*a*b*c^(3/2)*x*sgn(x))/c^2)/c

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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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