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

Optimal. Leaf size=44 \[ \frac {c \sqrt {c x^2} \log (x)}{a x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a x} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \frac {c \sqrt {c x^2} \log (x)}{a x}-\frac {c \sqrt {c x^2} \log (a+b x)}{a x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(x^4*(a + b*x)),x]

[Out]

(c*Sqrt[c*x^2]*Log[x])/(a*x) - (c*Sqrt[c*x^2]*Log[a + b*x])/(a*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 27, normalized size = 0.61 \[ \frac {\left (c x^2\right )^{3/2} (\log (x)-\log (a+b x))}{a x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 66, normalized size = 1.50 \[ \left [\frac {\sqrt {c x^{2}} c \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} c \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[sqrt(c*x^2)*c*log(x/(b*x + a))/(a*x), 2*sqrt(-c)*c*arctan(sqrt(c*x^2)*(2*b*x + a)*sqrt(-c)/(a*c*x))/a]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^4/(b*x+a),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 = 0.00, size = 26, normalized size = 0.59 \[ \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (\ln \relax (x )-\ln \left (b x +a \right )\right )}{a \,x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.38, size = 24, normalized size = 0.55 \[ -\frac {c^{\frac {3}{2}} \log \left (b x + a\right )}{a} + \frac {c^{\frac {3}{2}} \log \relax (x)}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/(x^4*(a + b*x)),x)

[Out]

int((c*x^2)^(3/2)/(x^4*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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