3.9.64 \(\int \frac {(c x^2)^{3/2}}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {c \sqrt {c x^2}}{a x (a+b x)} \]

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Rubi [A]  time = 0.02, antiderivative size = 68, 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, 44} \begin {gather*} -\frac {c \sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {c \sqrt {c x^2} \log (x)}{a^2 x}+\frac {c \sqrt {c x^2}}{a x (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Sqrt[c*x^2])/(a*x*(a + b*x)) + (c*Sqrt[c*x^2]*Log[x])/(a^2*x) - (c*Sqrt[c*x^2]*Log[a + b*x])/(a^2*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 46, normalized size = 0.68 \begin {gather*} \frac {\left (c x^2\right )^{3/2} (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 x^3 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 48, normalized size = 0.71 \begin {gather*} \left (c x^2\right )^{3/2} \left (-\frac {\log (a+b x)}{a^2 x^3}+\frac {\log (x)}{a^2 x^3}+\frac {1}{a x^3 (a+b x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^2)^(3/2)*(1/(a*x^3*(a + b*x)) + Log[x]/(a^2*x^3) - Log[a + b*x]/(a^2*x^3))

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fricas [A]  time = 1.12, size = 47, normalized size = 0.69 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (a c + {\left (b c x + a c\right )} \log \left (\frac {x}{b x + a}\right )\right )}}{a^{2} b x^{2} + a^{3} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Sign error (%%%{a,0%%%}+%%%{b,1%%%})

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maple [A]  time = 0.01, size = 52, normalized size = 0.76 \begin {gather*} \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \ln \relax (x )-a \ln \left (b x +a \right )+a \right )}{\left (b x +a \right ) a^{2} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.40, size = 38, normalized size = 0.56 \begin {gather*} \frac {c^{\frac {3}{2}}}{a b x + a^{2}} - \frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{a^{2}} + \frac {c^{\frac {3}{2}} \log \relax (x)}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,x^2\right )}^{3/2}}{x^4\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*x^2)^(3/2)/(x^4*(a + b*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}}}{x^{4} \left (a + b x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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