Optimal. Leaf size=80 \[ \frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b}-\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 80, 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, 45}
\begin {gather*} -\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x}+\frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {c x^2}}{a+b x} \, dx &=\frac {\sqrt {c x^2} \int \frac {x^3}{a+b x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx}{x}\\ &=\frac {a^2 \sqrt {c x^2}}{b^3}-\frac {a x \sqrt {c x^2}}{2 b^2}+\frac {x^2 \sqrt {c x^2}}{3 b}-\frac {a^3 \sqrt {c x^2} \log (a+b x)}{b^4 x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 52, normalized size = 0.65 \begin {gather*} \frac {c x \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 52, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {c \,x^{2}}\, \left (-2 b^{3} x^{3}+3 a \,b^{2} x^{2}+6 a^{3} \ln \left (b x +a \right )-6 a^{2} b x \right )}{6 x \,b^{4}}\) | \(52\) |
risch | \(\frac {\sqrt {c \,x^{2}}\, \left (\frac {1}{3} b^{2} x^{3}-\frac {1}{2} a b \,x^{2}+a^{2} x \right )}{x \,b^{3}}-\frac {a^{3} \ln \left (b x +a \right ) \sqrt {c \,x^{2}}}{b^{4} x}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 110, normalized size = 1.38 \begin {gather*} -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a^{3} \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{4}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a^{3} \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{4}} - \frac {\sqrt {c x^{2}} a x}{2 \, b^{2}} + \frac {\sqrt {c x^{2}} a^{2}}{b^{3}} + \frac {\left (c x^{2}\right )^{\frac {3}{2}}}{3 \, b c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 51, normalized size = 0.64 \begin {gather*} \frac {{\left (2 \, b^{3} x^{3} - 3 \, a b^{2} x^{2} + 6 \, a^{2} b x - 6 \, a^{3} \log \left (b x + a\right )\right )} \sqrt {c x^{2}}}{6 \, b^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \sqrt {c x^{2}}}{a + b x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.93, size = 69, normalized size = 0.86 \begin {gather*} -\frac {1}{6} \, \sqrt {c} {\left (\frac {6 \, a^{3} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {6 \, a^{3} \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right )}{b^{4}} - \frac {2 \, b^{2} x^{3} \mathrm {sgn}\left (x\right ) - 3 \, a b x^{2} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} x \mathrm {sgn}\left (x\right )}{b^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\sqrt {c\,x^2}}{a+b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________