Optimal. Leaf size=398 \[ \frac {4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 d^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 b d^2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b c d^2 x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {d^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.39, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {4763, 4857,
3398, 3377, 2718, 3392, 32, 2715, 8} \begin {gather*} \frac {d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^3}{2 b c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 d^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2}{c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {d^2 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 {b c d^2 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 {4 b d^2 x \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{\sqrt {c d x+d} \sqrt {e-c e x}}-\frac {b^2 d^2 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 32
Rule 2715
Rule 2718
Rule 3377
Rule 3392
Rule 3398
Rule 4763
Rule 4857
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {e-c e x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(d+c d x)^2 \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 d+c d \sin (x))^2 \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (c^2 d^2 (a+b x)^2+2 c^2 d^2 (a+b x)^2 \sin (x)+c^2 d^2 (a+b x)^2 \sin ^2(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {d^2 \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 (d^2 \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 {\left (2 d^2 \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 {b c d^2 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 d^2 \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 {d^2 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 {d^2 \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 (d^2 \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 (4 b d^2 \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 (b^2 d^2 \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 {b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 b d^2 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 {b c d^2 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 d^2 \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 {d^2 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 {d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (b^2 d^2 \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 (4 b^2 d^2 \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 {4 b^2 d^2 \left (1-c^2 x^2\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {b^2 d^2 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {b^2 d^2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {4 b d^2 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 {b c d^2 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 d^2 \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 {d^2 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 {d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.23, size = 344, normalized size = 0.86 \begin {gather*} \frac {b d \sqrt {d+c d x} \sqrt {e-c e x} \left (-4 a (4+c x) \sqrt {1-c^2 x^2}+b \left (-1+16 c x+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)-2 b d \sqrt {d+c d x} \sqrt {e-c e x} \left (-3 a+b (4+c x) \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2+2 b^2 d \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-6 a^2 d^{3/2} \sqrt {e} \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 )+d \sqrt {d+c d x} \sqrt {e-c e x} \left (16 a b c x-2 a^2 (4+c x) \sqrt {1-c^2 x^2}+b^2 (16+c x) \sqrt {1-c^2 x^2}-a b \cos (2 \text {ArcSin}(c x))\right )}{4 c e \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.17, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {-c e x +e}}\, dx\]
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\sqrt {- e \left (c x - 1\right )}}\, dx \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 = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{3/2}}{\sqrt {e-c\,e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________