Optimal. Leaf size=163 \[ -\frac {2 b f^3 \left (1-c^2 x^2\right )^{5/2}}{3 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {b f^3 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4763, 665,
4845, 12, 641, 45} \begin {gather*} -\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {2 b f^3 \left (1-c^2 x^2\right )^{5/2}}{3 c (c x+1) (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b f^3 \left (1-c^2 x^2\right )^{5/2} \log (c x+1)}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 641
Rule 665
Rule 4763
Rule 4845
Rubi steps
\begin {align*} \int \frac {\sqrt {f-c f x} \left (a+b \sin ^{-1}(c x)\right )}{(d+c d x)^{5/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{5/2} \int \frac {(f-c f x)^3 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{5/2}\right ) \int -\frac {f^3 (1-c x)^3}{3 c \left (1-c^2 x^2\right )^2} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(1-c x)^3}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1-c x}{(1+c x)^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (b f^3 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (\frac {1}{-1-c x}+\frac {2}{(1+c x)^2}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {2 b f^3 \left (1-c^2 x^2\right )^{5/2}}{3 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {f^3 (1-c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {b f^3 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 114, normalized size = 0.70 \begin {gather*} -\frac {f \sqrt {d+c d x} \left ((-1+c x) \left (-a+a c x-b \sqrt {1-c^2 x^2}\right )+b (-1+c x)^2 \text {ArcSin}(c x)+b (1+c x) \sqrt {1-c^2 x^2} \log (-f (1+c x))\right )}{3 c d^3 (1+c x)^2 \sqrt {f-c f x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arcsin \left (c x \right )\right ) \sqrt {-c f x +f}}{\left (c d x +d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 215, normalized size = 1.32 \begin {gather*} -\frac {1}{3} \, b c {\left (\frac {2 \, \sqrt {f}}{c^{3} d^{\frac {5}{2}} x + c^{2} d^{\frac {5}{2}}} + \frac {\sqrt {f} \log \left (c x + 1\right )}{c^{2} d^{\frac {5}{2}}}\right )} - \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {-c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}} - \frac {\sqrt {-c^{2} d f x^{2} + d f}}{c^{2} d^{3} x + c d^{3}}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {-c^{2} d f x^{2} + d f}}{c^{3} d^{3} x^{2} + 2 \, c^{2} d^{3} x + c d^{3}} - \frac {\sqrt {-c^{2} d f x^{2} + d f}}{c^{2} d^{3} x + c d^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.50, size = 520, normalized size = 3.19 \begin {gather*} \left [\frac {{\left (b c^{3} d x^{3} + b c^{2} d x^{2} - b c d x - b d\right )} \sqrt {\frac {f}{d}} \log \left (\frac {c^{6} f x^{6} + 4 \, c^{5} f x^{5} + 5 \, c^{4} f x^{4} - 4 \, c^{2} f x^{2} - 4 \, c f x + {\left (c^{4} x^{4} + 4 \, c^{3} x^{3} + 6 \, c^{2} x^{2} + 4 \, c x\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {\frac {f}{d}} - 2 \, f}{c^{4} x^{4} + 2 \, c^{3} x^{3} - 2 \, c x - 1}\right ) + 2 \, {\left (a c^{2} x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b c x - 2 \, a c x + {\left (b c^{2} x^{2} - 2 \, b c x + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {c d x + d} \sqrt {-c f x + f}}{6 \, {\left (c^{4} d^{3} x^{3} + c^{3} d^{3} x^{2} - c^{2} d^{3} x - c d^{3}\right )}}, -\frac {{\left (b c^{3} d x^{3} + b c^{2} d x^{2} - b c d x - b d\right )} \sqrt {-\frac {f}{d}} \arctan \left (\frac {{\left (c^{2} x^{2} + 2 \, c x + 2\right )} \sqrt {-c^{2} x^{2} + 1} \sqrt {c d x + d} \sqrt {-c f x + f} \sqrt {-\frac {f}{d}}}{c^{4} f x^{4} + 2 \, c^{3} f x^{3} - c^{2} f x^{2} - 2 \, c f x}\right ) - {\left (a c^{2} x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b c x - 2 \, a c x + {\left (b c^{2} x^{2} - 2 \, b c x + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt {c d x + d} \sqrt {-c f x + f}}{3 \, {\left (c^{4} d^{3} x^{3} + c^{3} d^{3} x^{2} - c^{2} d^{3} x - c d^{3}\right )}}\right ] \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 {\sqrt {- f \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d \left (c x + 1\right )\right )^{\frac {5}{2}}}\, 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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {f-c\,f\,x}}{{\left (d+c\,d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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