Optimal. Leaf size=33 \[ \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (1+p)}}{2 a (1+p)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37}
\begin {gather*} \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (p+1)}}{2 a (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 37
Rubi steps
\begin {align*} \int x \left (c x^2\right )^p (a+b x)^{-3-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{1+2 p} (a+b x)^{-3-2 p} \, dx\\ &=\frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (1+p)}}{2 a (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 32, normalized size = 0.97 \begin {gather*} \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2-2 p}}{a (2+2 p)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 32, normalized size = 0.97
method | result | size |
gosper | \(\frac {x^{2} \left (b x +a \right )^{-2-2 p} \left (c \,x^{2}\right )^{p}}{2 a \left (1+p \right )}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.87, size = 38, normalized size = 1.15 \begin {gather*} \frac {{\left (b x^{3} + a x^{2}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 3}}{2 \, {\left (a p + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1409 vs.
\(2 (27) = 54\).
time = 145.45, size = 1409, normalized size = 42.70 \begin {gather*} \begin {cases} - \frac {\left (b x\right )^{- 2 p} \left (c x^{2}\right )^{p}}{b^{3} x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 3} x^{2} \left (c x^{2}\right )^{p}}{2 p + 2} & \text {for}\: a = - b x \\\frac {x^{2} \left (c x^{2}\right )^{p} \left (0^{\frac {1}{p}}\right )^{- 2 p - 3}}{2 p + 2} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\frac {\frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a}}{c} & \text {for}\: p = -1 \\\frac {a^{4} x^{2} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {3 a^{3} b x^{3} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {3 a^{2} b^{2} x^{4} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {a b^{3} x^{5} \left (c x^{2}\right )^{p}}{2 a^{7} p \left (a + b x\right )^{2 p} + 2 a^{7} \left (a + b x\right )^{2 p} + 12 a^{6} b p x \left (a + b x\right )^{2 p} + 12 a^{6} b x \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 30 a^{5} b^{2} x^{2} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 40 a^{4} b^{3} x^{3} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} p x^{4} \left (a + b x\right )^{2 p} + 30 a^{3} b^{4} x^{4} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} p x^{5} \left (a + b x\right )^{2 p} + 12 a^{2} b^{5} x^{5} \left (a + b x\right )^{2 p} + 2 a b^{6} p x^{6} \left (a + b x\right )^{2 p} + 2 a b^{6} x^{6} \left (a + b x\right )^{2 p}} + \frac {b x^{3} \left (c x^{2}\right )^{p}}{2 a^{4} p \left (a + b x\right )^{2 p} + 2 a^{4} \left (a + b x\right )^{2 p} + 6 a^{3} b p x \left (a + b x\right )^{2 p} + 6 a^{3} b x \left (a + b x\right )^{2 p} + 6 a^{2} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{2} b^{2} x^{2} \left (a + b x\right )^{2 p} + 2 a b^{3} p x^{3} \left (a + b x\right )^{2 p} + 2 a b^{3} x^{3} \left (a + b x\right )^{2 p}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (33) = 66\).
time = 0.79, size = 72, normalized size = 2.18 \begin {gather*} \frac {\left (c x^{2}\right )^{p} b x^{3} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )} + \left (c x^{2}\right )^{p} a x^{2} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )}}{2 \, {\left (a p + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.22, size = 33, normalized size = 1.00 \begin {gather*} \frac {x^2\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+1\right )\,{\left (a+b\,x\right )}^{2\,p+2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________