Optimal. Leaf size=29 \[ \frac {b (c x)^{4+m} \sqrt {b x^2}}{c^4 (4+m) x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 16, 32}
\begin {gather*} \frac {b \sqrt {b x^2} (c x)^{m+4}}{c^4 (m+4) x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 16
Rule 32
Rubi steps
\begin {align*} \int (c x)^m \left (b x^2\right )^{3/2} \, dx &=\frac {\left (b \sqrt {b x^2}\right ) \int x^3 (c x)^m \, dx}{x}\\ &=\frac {\left (b \sqrt {b x^2}\right ) \int (c x)^{3+m} \, dx}{c^3 x}\\ &=\frac {b (c x)^{4+m} \sqrt {b x^2}}{c^4 (4+m) x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 21, normalized size = 0.72 \begin {gather*} \frac {x (c x)^m \left (b x^2\right )^{3/2}}{4+m} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 20, normalized size = 0.69
method | result | size |
gosper | \(\frac {x \left (c x \right )^{m} \left (b \,x^{2}\right )^{\frac {3}{2}}}{4+m}\) | \(20\) |
risch | \(\frac {b \,x^{3} \sqrt {b \,x^{2}}\, \left (c x \right )^{m}}{4+m}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 18, normalized size = 0.62 \begin {gather*} \frac {b^{\frac {3}{2}} c^{m} x^{4} x^{m}}{m + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 22, normalized size = 0.76 \begin {gather*} \frac {\sqrt {b x^{2}} \left (c x\right )^{m} b x^{3}}{m + 4} \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*} \begin {cases} \frac {x \left (b x^{2}\right )^{\frac {3}{2}} \left (c x\right )^{m}}{m + 4} & \text {for}\: m \neq -4 \\\frac {\int \frac {\left (b x^{2}\right )^{\frac {3}{2}}}{x^{4}}\, dx}{c^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.96, size = 22, normalized size = 0.76 \begin {gather*} \frac {b^{3/2}\,x^3\,{\left (c\,x\right )}^m\,\sqrt {x^2}}{m+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________