Optimal. Leaf size=345 \[ \frac {11 b d^3 x \sqrt {1-c^2 x^2}}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c^2 d^3 x^3 \sqrt {1-c^2 x^2}}{9 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{2 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.40, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {4763, 4847,
4737, 4767, 8, 4795, 30} \begin {gather*} \frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 b c \sqrt {c d x+d} \sqrt {f-c f x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 \sqrt {c d x+d} \sqrt {f-c f x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{2 \sqrt {c d x+d} \sqrt {f-c f x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c \sqrt {c d x+d} \sqrt {f-c f x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {c d x+d} \sqrt {f-c f x}}+\frac {11 b d^3 x \sqrt {1-c^2 x^2}}{3 \sqrt {c d x+d} \sqrt {f-c f x}}+\frac {b c^2 d^3 x^3 \sqrt {1-c^2 x^2}}{9 \sqrt {c d x+d} \sqrt {f-c f x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 4737
Rule 4763
Rule 4767
Rule 4795
Rule 4847
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {f-c f x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(d+c d x)^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}\\ &=\frac {\sqrt {1-c^2 x^2} \int \left (\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 c d^3 x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 c^2 d^3 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {c^3 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\right ) \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}\\ &=\frac {\left (d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (3 c d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (3 c^2 d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (c^3 d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}\\ &=-\frac {3 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (3 d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (3 b d^3 \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (2 c d^3 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (3 b c d^3 \sqrt {1-c^2 x^2}\right ) \int x \, dx}{2 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (b c^2 d^3 \sqrt {1-c^2 x^2}\right ) \int x^2 \, dx}{3 \sqrt {d+c d x} \sqrt {f-c f x}}\\ &=\frac {3 b d^3 x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c^2 d^3 x^3 \sqrt {1-c^2 x^2}}{9 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (2 b d^3 \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{3 \sqrt {d+c d x} \sqrt {f-c f x}}\\ &=\frac {11 b d^3 x \sqrt {1-c^2 x^2}}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c^2 d^3 x^3 \sqrt {1-c^2 x^2}}{9 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.27, size = 270, normalized size = 0.78 \begin {gather*} -\frac {d^2 \left (-90 b \sqrt {d+c d x} \sqrt {f-c f x} \text {ArcSin}(c x)^2+180 a \sqrt {d} \sqrt {f} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {f-c f x} \text {ArcSin}(c x) \left (9 (5+2 c x) \sqrt {1-c^2 x^2}-\cos (3 \text {ArcSin}(c x))\right )+\sqrt {d+c d x} \sqrt {f-c f x} \left (-270 b c x+12 a \sqrt {1-c^2 x^2} \left (22+9 c x+2 c^2 x^2\right )+27 b \cos (2 \text {ArcSin}(c x))+2 b \sin (3 \text {ArcSin}(c x))\right )\right )}{72 c f \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\sqrt {-c f x +f}}\, dx\]
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(-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.
[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 )\,{\left (d+c\,d\,x\right )}^{5/2}}{\sqrt {f-c\,f\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________