Optimal. Leaf size=61 \[ \frac {2 b x \sqrt {1-c^2 x^4}}{9 c}+\frac {1}{3} x^3 \left (a+b \text {ArcSin}\left (c x^2\right )\right )-\frac {2 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4926, 12, 327,
227} \begin {gather*} \frac {1}{3} x^3 \left (a+b \text {ArcSin}\left (c x^2\right )\right )-\frac {2 b F\left (\left .\text {ArcSin}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}}+\frac {2 b x \sqrt {1-c^2 x^4}}{9 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 227
Rule 327
Rule 4926
Rubi steps
\begin {align*} \int x^2 \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{3} b \int \frac {2 c x^4}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {1}{3} (2 b c) \int \frac {x^4}{\sqrt {1-c^2 x^4}} \, dx\\ &=\frac {2 b x \sqrt {1-c^2 x^4}}{9 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {(2 b) \int \frac {1}{\sqrt {1-c^2 x^4}} \, dx}{9 c}\\ &=\frac {2 b x \sqrt {1-c^2 x^4}}{9 c}+\frac {1}{3} x^3 \left (a+b \sin ^{-1}\left (c x^2\right )\right )-\frac {2 b F\left (\left .\sin ^{-1}\left (\sqrt {c} x\right )\right |-1\right )}{9 c^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 72, normalized size = 1.18 \begin {gather*} \frac {1}{9} \left (3 a x^3+\frac {2 b x \sqrt {1-c^2 x^4}}{c}+3 b x^3 \text {ArcSin}\left (c x^2\right )-\frac {2 i b F\left (\left .i \sinh ^{-1}\left (\sqrt {-c} x\right )\right |-1\right )}{(-c)^{3/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.01, size = 88, normalized size = 1.44
method | result | size |
default | \(\frac {x^{3} a}{3}+b \left (\frac {x^{3} \arcsin \left (c \,x^{2}\right )}{3}-\frac {2 c \left (-\frac {x \sqrt {-c^{2} x^{4}+1}}{3 c^{2}}+\frac {\sqrt {-c \,x^{2}+1}\, \sqrt {c \,x^{2}+1}\, \EllipticF \left (x \sqrt {c}, i\right )}{3 c^{\frac {5}{2}} \sqrt {-c^{2} x^{4}+1}}\right )}{3}\right )\) | \(88\) |
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 = 0.45, size = 42, normalized size = 0.69 \begin {gather*} \frac {3 \, b c x^{3} \arcsin \left (c x^{2}\right ) + 3 \, a c x^{3} + 2 \, \sqrt {-c^{2} x^{4} + 1} b x}{9 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.09, size = 58, normalized size = 0.95 \begin {gather*} \frac {a x^{3}}{3} - \frac {b c x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {c^{2} x^{4} e^{2 i \pi }} \right )}}{6 \Gamma \left (\frac {9}{4}\right )} + \frac {b x^{3} \operatorname {asin}{\left (c x^{2} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x^2\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________