3.9.19 \(\int \frac {(c x^2)^{3/2} (a+b x)^2}{x^4} \, dx\) [819]

Optimal. Leaf size=52 \[ 2 a b c \sqrt {c x^2}+\frac {1}{2} b^2 c x \sqrt {c x^2}+\frac {a^2 c \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 52, 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 c \sqrt {c x^2} \log (x)}{x}+2 a b c \sqrt {c x^2}+\frac {1}{2} b^2 c x \sqrt {c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a*b*c*Sqrt[c*x^2] + (b^2*c*x*Sqrt[c*x^2])/2 + (a^2*c*Sqrt[c*x^2]*Log[x])/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 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 {\left (c x^2\right )^{3/2} (a+b x)^2}{x^4} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int \frac {(a+b x)^2}{x} \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b c \sqrt {c x^2}+\frac {1}{2} b^2 c x \sqrt {c x^2}+\frac {a^2 c \sqrt {c x^2} \log (x)}{x}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.65 \begin {gather*} \frac {\left (c x^2\right )^{3/2} \left (b x (4 a+b x)+2 a^2 \log (x)\right )}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[((c*x^2)^(3/2)*(a + b*x)^2)/x^4,x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.10, size = 33, normalized size = 0.63

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)*(b*x+a)^2/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.30, size = 35, normalized size = 0.67 \begin {gather*} \frac {{\left (b^{2} c x^{2} + 4 \, a b c x + 2 \, a^{2} c \log \left (x\right )\right )} \sqrt {c x^{2}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)^2/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)*(b*x+a)**2/x**4,x)

[Out]

Integral((c*x**2)**(3/2)*(a + b*x)**2/x**4, x)

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Giac [A]
time = 0.00, size = 35, normalized size = 0.67 \begin {gather*} \sqrt {c} c \left (a^{2} \mathrm {sign}\left (x\right ) \ln \left |x\right |+\frac {1}{2} b^{2} x^{2} \mathrm {sign}\left (x\right )+2 a b x \mathrm {sign}\left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)*(b*x+a)^2/x^4,x)

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*x^2)^(3/2)*(a + b*x)^2)/x^4,x)

[Out]

int(((c*x^2)^(3/2)*(a + b*x)^2)/x^4, x)

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