Optimal. Leaf size=559 \[ -\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]
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Rubi [A]
time = 0.48, antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4763, 4857,
3398, 3377, 2718, 3392, 32, 2715, 8, 2713} \begin {gather*} \frac {5 e^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{3 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{9 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 32
Rule 2713
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 4763
Rule 4857
Rubi steps
\begin {align*} \int \frac {(e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {d+c d x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(e-c e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 (c e-c e \sin (x))^3 \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (c^3 e^3 (a+b x)^2-3 c^3 e^3 (a+b x)^2 \sin (x)+3 c^3 e^3 (a+b x)^2 \sin ^2(x)-c^3 e^3 (a+b x)^2 \sin ^3(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (6 b e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {6 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (4 b e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (6 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {56 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (4 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
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Mathematica [A]
time = 1.89, size = 473, normalized size = 0.85 \begin {gather*} \frac {180 b^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-540 a^2 \sqrt {d} e^{5/2} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-6 b e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) \left (264 b c x+8 b c^3 x^3-270 a \sqrt {1-c^2 x^2}+108 a c x \sqrt {1-c^2 x^2}+27 b \cos (2 \text {ArcSin}(c x))+6 a \cos (3 \text {ArcSin}(c x))\right )+18 b e^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 \left (30 a+45 b \sqrt {1-c^2 x^2}-b \cos (3 \text {ArcSin}(c x))-9 b \sin (2 \text {ArcSin}(c x))\right )+e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (-1620 a b c x+792 a^2 \sqrt {1-c^2 x^2}-1620 b^2 \sqrt {1-c^2 x^2}-324 a^2 c x \sqrt {1-c^2 x^2}+72 a^2 c^2 x^2 \sqrt {1-c^2 x^2}-162 a b \cos (2 \text {ArcSin}(c x))+4 b^2 \cos (3 \text {ArcSin}(c x))+81 b^2 \sin (2 \text {ArcSin}(c x))+12 a b \sin (3 \text {ArcSin}(c x))\right )}{216 c d \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [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.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{5/2}}{\sqrt {d+c\,d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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