Optimal. Leaf size=242 \[ -\frac {2 b f^2 x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c f^2 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {2 f^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{c \sqrt {d+c d x} \sqrt {f-c f x}}-\frac {f^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{2 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {3 f^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{4 b c \sqrt {d+c d x} \sqrt {f-c f x}} \]
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Rubi [A]
time = 0.30, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 {3 f^2 \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 {f^2 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 {2 f^2 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{c \sqrt {c d x+d} \sqrt {f-c f x}}+\frac {b c f^2 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {c d x+d} \sqrt {f-c f x}}-\frac {2 b f^2 x \sqrt {1-c^2 x^2}}{\sqrt {c d x+d} \sqrt {f-c f x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 4737
Rule 4763
Rule 4767
Rule 4795
Rule 4847
Rubi steps
\begin {align*} \int \frac {(f-c f x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d+c d x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f-c f 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 {\sqrt {1-c^2 x^2} \int \left (\frac {f^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {2 c f^2 x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {c^2 f^2 x^2 \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 (f^2 \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 (2 c f^2 \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 (c^2 f^2 \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 {2 f^2 \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 {f^2 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 {f^2 \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 (f^2 \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 (2 b f^2 \sqrt {1-c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {\left (b c f^2 \sqrt {1-c^2 x^2}\right ) \int x \, dx}{2 \sqrt {d+c d x} \sqrt {f-c f x}}\\ &=-\frac {2 b f^2 x \sqrt {1-c^2 x^2}}{\sqrt {d+c d x} \sqrt {f-c f x}}+\frac {b c f^2 x^2 \sqrt {1-c^2 x^2}}{4 \sqrt {d+c d x} \sqrt {f-c f x}}+\frac {2 f^2 \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 {f^2 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 {3 f^2 \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*}
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Mathematica [A]
time = 0.65, size = 238, normalized size = 0.98 \begin {gather*} \frac {-4 b f (-4+c x) \sqrt {d+c d x} \sqrt {f-c f x} \sqrt {1-c^2 x^2} \text {ArcSin}(c x)+6 b f \sqrt {d+c d x} \sqrt {f-c f x} \text {ArcSin}(c x)^2-12 a \sqrt {d} f^{3/2} \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 )-f \sqrt {d+c d x} \sqrt {f-c f x} \left (16 b c x+4 a (-4+c x) \sqrt {1-c^2 x^2}+b \cos (2 \text {ArcSin}(c x))\right )}{8 c d \sqrt {1-c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (-c f x +f \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\sqrt {c d x +d}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- f \left (c x - 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\sqrt {d \left (c x + 1\right )}}\, dx \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 )\,{\left (f-c\,f\,x\right )}^{3/2}}{\sqrt {d+c\,d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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