3.8.96 \(\int \frac {x^3 (a+b x)}{(c x^2)^{5/2}} \, dx\) [796]

Optimal. Leaf size=33 \[ -\frac {a}{c^2 \sqrt {c x^2}}+\frac {b x \log (x)}{c^2 \sqrt {c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, 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, 45} \begin {gather*} \frac {b x \log (x)}{c^2 \sqrt {c x^2}}-\frac {a}{c^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^3*(a + b*x))/(c*x^2)^(5/2),x]

[Out]

-(a/(c^2*Sqrt[c*x^2])) + (b*x*Log[x])/(c^2*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^3 (a+b x)}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^2} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a}{c^2 \sqrt {c x^2}}+\frac {b x \log (x)}{c^2 \sqrt {c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 22, normalized size = 0.67 \begin {gather*} \frac {-a+b x \log (x)}{c^2 \sqrt {c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^3*(a + b*x))/(c*x^2)^(5/2),x]

[Out]

(-a + b*x*Log[x])/(c^2*Sqrt[c*x^2])

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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[(x^3*(a + b*x))/(c*x^2)^(5/2),x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.02, size = 21, normalized size = 0.64

method	result	size
default	\(\frac {x^{4} \left (b x \ln \left (x \right )-a \right )}{\left (c \,x^{2}\right )^{\frac {5}{2}}}\)	\(21\)
risch	\(-\frac {a}{c^{2} \sqrt {c \,x^{2}}}+\frac {b x \ln \left (x \right )}{c^{2} \sqrt {c \,x^{2}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)/(c*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

x^4*(b*x*ln(x)-a)/(c*x^2)^(5/2)

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Maxima [A]
time = 0.30, size = 24, normalized size = 0.73 \begin {gather*} -\frac {a x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} + \frac {b \log \left (x\right )}{c^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)/(c*x^2)^(5/2),x, algorithm="maxima")

[Out]

-a*x^2/((c*x^2)^(3/2)*c) + b*log(x)/c^(5/2)

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Fricas [A]
time = 0.30, size = 23, normalized size = 0.70 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x \log \left (x\right ) - a\right )}}{c^{3} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)/(c*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)/(c*x**2)**(5/2),x)

[Out]

Integral(x**3*(a + b*x)/(c*x**2)**(5/2), x)

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Giac [A]
time = 0.00, size = 31, normalized size = 0.94 \begin {gather*} \frac {-\frac {a}{c^{2} x \mathrm {sign}\left (x\right )}+\frac {b \ln \left |x\right |}{c^{2} \mathrm {sign}\left (x\right )}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)/(c*x^2)^(5/2),x)

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^3\,\left (a+b\,x\right )}{{\left (c\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a + b*x))/(c*x^2)^(5/2),x)

[Out]

int((x^3*(a + b*x))/(c*x^2)^(5/2), x)

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