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

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

[Out]

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

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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} \[ -\frac {x \log (a+b x)}{a^2 c \sqrt {c x^2}}+\frac {x \log (x)}{a^2 c \sqrt {c x^2}}+\frac {x}{a c \sqrt {c x^2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 46, normalized size = 0.68 \[ \frac {x^3 (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \left (c x^2\right )^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 48, normalized size = 0.71 \[ \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b c^{2} x^{2} + a^{3} c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.06, size = 83, normalized size = 1.22 \[ -\frac {\frac {\log \left ({\left | -\frac {a}{b x + a} + 1 \right |}\right )}{a^{2} \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )} + \frac {1}{{\left (b x + a\right )} a \mathrm {sgn}\left (-\frac {b}{b x + a} + \frac {a b}{{\left (b x + a\right )}^{2}}\right )}}{c^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.52, size = 82, normalized size = 1.21 \[ -\frac {1}{\sqrt {c x^{2}} b^{2} c x + \sqrt {c x^{2}} a b c} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{a^{2} c^{\frac {3}{2}}} + \frac {1}{\sqrt {c x^{2}} a b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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