3.892 \(\int \frac{1}{x (c x^2)^{3/2} (a+b x)} \, dx\)

Optimal. Leaf size=115 \[ -\frac{b^2}{a^3 c \sqrt{c x^2}}-\frac{b^3 x \log (x)}{a^4 c \sqrt{c x^2}}+\frac{b^3 x \log (a+b x)}{a^4 c \sqrt{c x^2}}+\frac{b}{2 a^2 c x \sqrt{c x^2}}-\frac{1}{3 a c x^2 \sqrt{c x^2}} \]

[Out]

-(b^2/(a^3*c*Sqrt[c*x^2])) - 1/(3*a*c*x^2*Sqrt[c*x^2]) + b/(2*a^2*c*x*Sqrt[c*x^2]) - (b^3*x*Log[x])/(a^4*c*Sqr
t[c*x^2]) + (b^3*x*Log[a + b*x])/(a^4*c*Sqrt[c*x^2])

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Rubi [A]  time = 0.0283939, antiderivative size = 115, 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} \[ -\frac{b^2}{a^3 c \sqrt{c x^2}}-\frac{b^3 x \log (x)}{a^4 c \sqrt{c x^2}}+\frac{b^3 x \log (a+b x)}{a^4 c \sqrt{c x^2}}+\frac{b}{2 a^2 c x \sqrt{c x^2}}-\frac{1}{3 a c x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

Mathematica [A]  time = 0.015498, size = 66, normalized size = 0.57 \[ \frac{c x^2 \left (a \left (-2 a^2+3 a b x-6 b^2 x^2\right )+6 b^3 x^3 \log (a+b x)-6 b^3 x^3 \log (x)\right )}{6 a^4 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(c*x^2)^(3/2)*(a + b*x)),x]

[Out]

(c*x^2*(a*(-2*a^2 + 3*a*b*x - 6*b^2*x^2) - 6*b^3*x^3*Log[x] + 6*b^3*x^3*Log[a + b*x]))/(6*a^4*(c*x^2)^(5/2))

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Maple [A]  time = 0.004, size = 59, normalized size = 0.5 \begin{align*} -{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,a{b}^{2}{x}^{2}-3\,{a}^{2}bx+2\,{a}^{3}}{6\,{a}^{4}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(c*x^2)^(3/2)/(b*x+a),x)

[Out]

-1/6*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*a*b^2*x^2-3*a^2*b*x+2*a^3)/(c*x^2)^(3/2)/a^4

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Maxima [A]  time = 1.04303, size = 93, normalized size = 0.81 \begin{align*} \frac{b^{3} \log \left (b x + a\right )}{a^{4} c^{\frac{3}{2}}} - \frac{b^{3} \log \left (x\right )}{a^{4} c^{\frac{3}{2}}} - \frac{6 \, b^{2} \sqrt{c} x^{2} - 3 \, a b \sqrt{c} x + 2 \, a^{2} \sqrt{c}}{6 \, a^{3} c^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^2)^(3/2)/(b*x+a),x, algorithm="maxima")

[Out]

b^3*log(b*x + a)/(a^4*c^(3/2)) - b^3*log(x)/(a^4*c^(3/2)) - 1/6*(6*b^2*sqrt(c)*x^2 - 3*a*b*sqrt(c)*x + 2*a^2*s
qrt(c))/(a^3*c^2*x^3)

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Fricas [A]  time = 1.60288, size = 127, normalized size = 1.1 \begin{align*} \frac{{\left (6 \, b^{3} x^{3} \log \left (\frac{b x + a}{x}\right ) - 6 \, a b^{2} x^{2} + 3 \, a^{2} b x - 2 \, a^{3}\right )} \sqrt{c x^{2}}}{6 \, a^{4} c^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^2)^(3/2)/(b*x+a),x, algorithm="fricas")

[Out]

1/6*(6*b^3*x^3*log((b*x + a)/x) - 6*a*b^2*x^2 + 3*a^2*b*x - 2*a^3)*sqrt(c*x^2)/(a^4*c^2*x^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x**2)**(3/2)/(b*x+a),x)

[Out]

Integral(1/(x*(c*x**2)**(3/2)*(a + b*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c x^{2}\right )^{\frac{3}{2}}{\left (b x + a\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(c*x^2)^(3/2)/(b*x+a),x, algorithm="giac")

[Out]

integrate(1/((c*x^2)^(3/2)*(b*x + a)*x), x)