Optimal. Leaf size=697 \[ -\frac {3 b c d x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {2 b d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {d (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {b d \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac {4 b c^2 d x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {1}{4} d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {2 b^2 d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}+\frac {16 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}+\frac {9 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{32} b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}+\frac {8 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.80, antiderivative size = 697, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {4673, 4763, 4649, 4647, 4641, 4627, 321, 216, 4677, 195, 194, 4645, 12, 1247, 698} \[ \frac {2 b c^4 d x^5 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c d x^2 (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}+\frac {3 d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}+\frac {2 b d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}+\frac {d (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {b d \sqrt {1-c^2 x^2} (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (c d x+d)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {15 b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {2 b^2 d \left (1-c^2 x^2\right ) (c d x+d)^{3/2} (e-c e x)^{3/2}}{125 c}+\frac {16 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}+\frac {9 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}-\frac {1}{32} b^2 d x (c d x+d)^{3/2} (e-c e x)^{3/2}+\frac {8 b^2 d (c d x+d)^{3/2} (e-c e x)^{3/2}}{225 c} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 194
Rule 195
Rule 216
Rule 321
Rule 698
Rule 1247
Rule 4627
Rule 4641
Rule 4645
Rule 4647
Rule 4649
Rule 4673
Rule 4677
Rule 4763
Rubi steps
\begin {align*} \int (d+c d x)^{5/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int (d+c d x) \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {\left ((d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (d \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+c d x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {\left (d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}+\frac {\left (c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {\left (3 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (2 b d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {\left (3 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (2 b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1-c^2 x^2}} \, dx}{5 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}+\frac {2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{32 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (2 b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx}{75 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 c^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (3 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \operatorname {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{75 \left (1-c^2 x^2\right )^{3/2}}\\ &=-\frac {1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}-\frac {15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {9 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}-\frac {\left (b^2 c d (d+c d x)^{3/2} (e-c e x)^{3/2}\right ) \operatorname {Subst}\left (\int \left (\frac {8}{\sqrt {1-c^2 x}}+4 \sqrt {1-c^2 x}+3 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \left (1-c^2 x^2\right )^{3/2}}\\ &=\frac {8 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}}{225 c}-\frac {1}{32} b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}+\frac {16 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2}}{75 c \left (1-c^2 x^2\right )}-\frac {15 b^2 d x (d+c d x)^{3/2} (e-c e x)^{3/2}}{64 \left (1-c^2 x^2\right )}+\frac {2 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right )}{125 c}+\frac {9 b^2 d (d+c d x)^{3/2} (e-c e x)^{3/2} \sin ^{-1}(c x)}{64 c \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b c d x^2 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{8 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^2 d x^3 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 \left (1-c^2 x^2\right )^{3/2}}+\frac {2 b c^4 d x^5 (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{25 \left (1-c^2 x^2\right )^{3/2}}+\frac {b d (d+c d x)^{3/2} (e-c e x)^{3/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac {1}{4} d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3 d x (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{8 \left (1-c^2 x^2\right )}-\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{5 c}+\frac {d (d+c d x)^{3/2} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^3}{8 b c \left (1-c^2 x^2\right )^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 4.04, size = 574, normalized size = 0.82 \[ \frac {d^2 e \left (\sqrt {c d x+d} \sqrt {e-c e x} \left (-15 \left (480 a^2 \sqrt {1-c^2 x^2} \left (8 c^4 x^4+10 c^3 x^3-16 c^2 x^2-25 c x+8\right )-512 a b c x \left (3 c^4 x^4-10 c^2 x^2+15\right )-4800 b^2 \sqrt {1-c^2 x^2}+2400 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+75 b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )+72000 a b \cos \left (2 \sin ^{-1}(c x)\right )+4500 a b \cos \left (4 \sin ^{-1}(c x)\right )+4000 b^2 \cos \left (3 \sin ^{-1}(c x)\right )+288 b^2 \cos \left (5 \sin ^{-1}(c x)\right )\right )-108000 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+1800 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (5 \left (12 a-4 b \sqrt {1-c^2 x^2}+8 b \sin \left (2 \sin ^{-1}(c x)\right )+b \sin \left (4 \sin ^{-1}(c x)\right )\right )-10 b \cos \left (3 \sin ^{-1}(c x)\right )-2 b \cos \left (5 \sin ^{-1}(c x)\right )\right )-60 b \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (-4 \left (960 a c^2 x^2 \sqrt {1-c^2 x^2}-480 a \sqrt {1-c^2 x^2}-480 a c^4 x^4 \sqrt {1-c^2 x^2}+600 a \sin \left (2 \sin ^{-1}(c x)\right )+75 a \sin \left (4 \sin ^{-1}(c x)\right )+300 b c x+50 b \sin \left (3 \sin ^{-1}(c x)\right )+6 b \sin \left (5 \sin ^{-1}(c x)\right )\right )-1200 b \cos \left (2 \sin ^{-1}(c x)\right )-75 b \cos \left (4 \sin ^{-1}(c x)\right )\right )+36000 b^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3\right )}{288000 c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c^{3} d^{2} e x^{3} + a^{2} c^{2} d^{2} e x^{2} - a^{2} c d^{2} e x - a^{2} d^{2} e + {\left (b^{2} c^{3} d^{2} e x^{3} + b^{2} c^{2} d^{2} e x^{2} - b^{2} c d^{2} e x - b^{2} d^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{3} d^{2} e x^{3} + a b c^{2} d^{2} e x^{2} - a b c d^{2} e x - a b d^{2} e\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (c d x +d \right )^{\frac {5}{2}} \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{40} \, {\left (15 \, \sqrt {-c^{2} d e x^{2} + d e} d^{2} e x + \frac {15 \, d^{3} e^{2} \arcsin \left (c x\right )}{\sqrt {d e} c} + 10 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} d x - \frac {8 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {5}{2}}}{c e}\right )} a^{2} + \sqrt {d} \sqrt {e} \int -{\left ({\left (b^{2} c^{3} d^{2} e x^{3} + b^{2} c^{2} d^{2} e x^{2} - b^{2} c d^{2} e x - b^{2} d^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{3} d^{2} e x^{3} + a b c^{2} d^{2} e x^{2} - a b c d^{2} e x - a b d^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________