Optimal. Leaf size=150 \[ -\frac {b x \sqrt {d-c^2 d x^2}}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {a+b \text {ArcSin}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {ArcSin}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {272, 45, 4779,
12, 393, 212} \begin {gather*} -\frac {a+b \text {ArcSin}(c x)}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {a+b \text {ArcSin}(c x)}{3 c^4 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 b \sqrt {d-c^2 d x^2} \tanh ^{-1}(c x)}{6 c^4 d^3 \sqrt {1-c^2 x^2}}-\frac {b x \sqrt {d-c^2 d x^2}}{6 c^3 d^3 \left (1-c^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 212
Rule 272
Rule 393
Rule 4779
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {x^2}{\left (1-c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{6 c^3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \int \frac {1}{1-c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b x}{6 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 b \sqrt {1-c^2 x^2} \tanh ^{-1}(c x)}{6 c^4 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.15, size = 143, normalized size = 0.95 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (\sqrt {-c^2} \left (-4 a+6 a c^2 x^2-b c x \sqrt {1-c^2 x^2}+2 b \left (-2+3 c^2 x^2\right ) \text {ArcSin}(c x)\right )-5 i b c \left (1-c^2 x^2\right )^{3/2} F\left (\left .i \sinh ^{-1}\left (\sqrt {-c^2} x\right )\right |1\right )\right )}{6 c^4 \sqrt {-c^2} d^3 \left (-1+c^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains complex when optimal does not.
time = 0.25, size = 308, normalized size = 2.05
method | result | size |
default | \(a \left (\frac {x^{2}}{c^{2} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x^{2}}{d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{2}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, x}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{3}}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 d^{3} \left (c^{2} x^{2}-1\right )^{2} c^{4}}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}+i\right )}{6 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 160, normalized size = 1.07 \begin {gather*} \frac {1}{12} \, b c {\left (\frac {2 \, x}{c^{6} d^{\frac {5}{2}} x^{2} - c^{4} d^{\frac {5}{2}}} + \frac {5 \, \log \left (c x + 1\right )}{c^{5} d^{\frac {5}{2}}} - \frac {5 \, \log \left (c x - 1\right )}{c^{5} d^{\frac {5}{2}}}\right )} + \frac {1}{3} \, b {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \arcsin \left (c x\right ) + \frac {1}{3} \, a {\left (\frac {3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d} - \frac {2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{4} d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.52, size = 421, normalized size = 2.81 \begin {gather*} \left [-\frac {4 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x - 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \log \left (-\frac {c^{6} d x^{6} + 5 \, c^{4} d x^{4} - 5 \, c^{2} d x^{2} - 4 \, {\left (c^{3} x^{3} + c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} \sqrt {d} - d}{c^{6} x^{6} - 3 \, c^{4} x^{4} + 3 \, c^{2} x^{2} - 1}\right ) - 8 \, {\left (3 \, a c^{2} x^{2} + {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{24 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}, -\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} b c x - 5 \, {\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \sqrt {-d} \arctan \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} c \sqrt {-d} x}{c^{4} d x^{4} - d}\right ) - 4 \, {\left (3 \, a c^{2} x^{2} + {\left (3 \, b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________