3.867 \(\int \frac{(c x^2)^{3/2}}{x^6 (a+b x)} \, dx\)

Optimal. Leaf size=88 \[ \frac{b^2 c \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 c \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b c \sqrt{c x^2}}{a^2 x^2}-\frac{c \sqrt{c x^2}}{2 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0236948, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 44} \[ \frac{b^2 c \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 c \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b c \sqrt{c x^2}}{a^2 x^2}-\frac{c \sqrt{c x^2}}{2 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(x^6*(a + b*x)),x]

[Out]

-(c*Sqrt[c*x^2])/(2*a*x^3) + (b*c*Sqrt[c*x^2])/(a^2*x^2) + (b^2*c*Sqrt[c*x^2]*Log[x])/(a^3*x) - (b^2*c*Sqrt[c*
x^2]*Log[a + b*x])/(a^3*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^6 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{x^3 (a+b x)} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{c \sqrt{c x^2}}{2 a x^3}+\frac{b c \sqrt{c x^2}}{a^2 x^2}+\frac{b^2 c \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 c \sqrt{c x^2} \log (a+b x)}{a^3 x}\\ \end{align*}

Mathematica [A]  time = 0.0142415, size = 53, normalized size = 0.6 \[ \frac{\left (c x^2\right )^{3/2} \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x^2)^(3/2)/(x^6*(a + b*x)),x]

[Out]

((c*x^2)^(3/2)*(-(a*(a - 2*b*x)) + 2*b^2*x^2*Log[x] - 2*b^2*x^2*Log[a + b*x]))/(2*a^3*x^5)

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Maple [A]  time = 0.003, size = 51, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,{a}^{3}{x}^{5}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/x^6/(b*x+a),x)

[Out]

1/2*(c*x^2)^(3/2)*(2*b^2*ln(x)*x^2-2*b^2*ln(b*x+a)*x^2+2*a*b*x-a^2)/x^5/a^3

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Maxima [A]  time = 1.11167, size = 70, normalized size = 0.8 \begin{align*} -\frac{b^{2} c^{\frac{3}{2}} \log \left (b x + a\right )}{a^{3}} + \frac{b^{2} c^{\frac{3}{2}} \log \left (x\right )}{a^{3}} + \frac{2 \, b c^{\frac{3}{2}} x - a c^{\frac{3}{2}}}{2 \, a^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.61361, size = 105, normalized size = 1.19 \begin{align*} \frac{{\left (2 \, b^{2} c x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b c x - a^{2} c\right )} \sqrt{c x^{2}}}{2 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*b^2*c*x^2*log(x/(b*x + a)) + 2*a*b*c*x - a^2*c)*sqrt(c*x^2)/(a^3*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{6} \left (a + b x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2)**(3/2)/x**6/(b*x+a),x)

[Out]

Integral((c*x**2)**(3/2)/(x**6*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2)^(3/2)/x^6/(b*x+a),x, algorithm="giac")

[Out]

Exception raised: TypeError