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

Optimal. Leaf size=40 \[ \frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 40, 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, 43} \[ \frac {c \sqrt {c x^2}}{b}-\frac {a c \sqrt {c x^2} \log (a+b x)}{b^2 x} \]

Antiderivative was successfully verified.

[In]

Int[(c*x^2)^(3/2)/(x^2*(a + b*x)),x]

[Out]

(c*Sqrt[c*x^2])/b - (a*c*Sqrt[c*x^2]*Log[a + b*x])/(b^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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 30, normalized size = 0.75 \[ \frac {c^2 x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 29, normalized size = 0.72 \[ \frac {{\left (b c x - a c \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{b^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 37, normalized size = 0.92 \[ c^{\frac {3}{2}} {\left (\frac {x \mathrm {sgn}\relax (x)}{b} - \frac {a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}} + \frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 29, normalized size = 0.72 \[ -\frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.48, size = 75, normalized size = 1.88 \[ -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a c^{\frac {3}{2}} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a c^{\frac {3}{2}} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}} c}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2)^(3/2)/(x^2*(a + b*x)),x)

[Out]

int((c*x^2)^(3/2)/(x^2*(a + b*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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