Optimal. Leaf size=463 \[ \frac {5 d^4 (c x+1) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (c x+1)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
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Rubi [A] time = 0.37, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4673, 669, 671, 641, 216, 4761, 627, 43, 4641} \[ \frac {5 d^4 (c x+1) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (c x+1)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (c x+1)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}}-\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (c d x+d)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (c d x+d)^{3/2} (f-c f x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 216
Rule 627
Rule 641
Rule 669
Rule 671
Rule 4641
Rule 4673
Rule 4761
Rubi steps
\begin {align*} \int \frac {(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{(f-c f x)^{3/2}} \, dx &=\frac {\left (1-c^2 x^2\right )^{3/2} \int \frac {(d+c d x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (b c \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (\frac {15 d^4}{2 c}+\frac {5 d^4 (1+c x)}{2 c}+\frac {2 d^4 (1+c x)^3}{c \left (1-c^2 x^2\right )}-\frac {15 d^4 \sin ^{-1}(c x)}{2 c \sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(1+c x)^3}{1-c^2 x^2} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {\left (15 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \frac {(1+c x)^2}{1-c x} \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {15 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {\left (2 b d^4 \left (1-c^2 x^2\right )^{3/2}\right ) \int \left (-3-c x+\frac {4}{1-c x}\right ) \, dx}{(d+c d x)^{3/2} (f-c f x)^{3/2}}\\ &=-\frac {3 b d^4 x \left (1-c^2 x^2\right )^{3/2}}{2 (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {b c d^4 x^2 \left (1-c^2 x^2\right )^{3/2}}{(d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {5 b d^4 (1+c x)^2 \left (1-c^2 x^2\right )^{3/2}}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 b d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x)^2}{4 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {2 d^4 (1+c x)^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {15 d^4 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {5 d^4 (1+c x) \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}-\frac {15 d^4 \left (1-c^2 x^2\right )^{3/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{2 c (d+c d x)^{3/2} (f-c f x)^{3/2}}+\frac {8 b d^4 \left (1-c^2 x^2\right )^{3/2} \log (1-c x)}{c (d+c d x)^{3/2} (f-c f x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 4.55, size = 768, normalized size = 1.66 \[ \frac {d^2 \left (\frac {8 a \left (c^2 x^2+7 c x-24\right ) \sqrt {c d x+d} \sqrt {f-c f x}}{c x-1}+120 a \sqrt {d} \sqrt {f} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {f-c f x}}{\sqrt {d} \sqrt {f} \left (c^2 x^2-1\right )}\right )-\frac {8 b (c x+1) \sqrt {c d x+d} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (\left (\sin ^{-1}(c x)-4\right ) \sin ^{-1}(c x)-8 \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right ) \left (\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+4\right )-8 \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^2}-\frac {32 b (c x+1) \sqrt {c d x+d} \sqrt {f-c f x} \left (-\sin ^{-1}(c x) \left (\left (\sqrt {1-c^2 x^2}+2\right ) \cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\left (\sqrt {1-c^2 x^2}-2\right ) \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+\sin ^{-1}(c x)^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+\left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (c x-4 \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^2}+\frac {b (c x+1) \sqrt {c d x+d} \sqrt {f-c f x} \left (-20 \sin ^{-1}(c x)^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )+2 \sin ^{-1}(c x) \left (24 \sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )-7 \sin \left (\frac {3}{2} \sin ^{-1}(c x)\right )+\sin \left (\frac {5}{2} \sin ^{-1}(c x)\right )+24 \cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )+7 \cos \left (\frac {3}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {5}{2} \sin ^{-1}(c x)\right )\right )+2 \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (-16 c x+\cos \left (2 \sin ^{-1}(c x)\right )+32 \log \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )\right )\right )}{\sqrt {1-c^2 x^2} \left (\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right ) \left (\sin \left (\frac {1}{2} \sin ^{-1}(c x)\right )+\cos \left (\frac {1}{2} \sin ^{-1}(c x)\right )\right )^2}\right )}{16 c f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a c^{2} d^{2} x^{2} + 2 \, a c d^{2} x + a d^{2} + {\left (b c^{2} d^{2} x^{2} + 2 \, b c d^{2} x + b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c f x + f}}{c^{2} f^{2} x^{2} - 2 \, c f^{2} x + f^{2}}, x\right ) \]
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 \[ \int \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (-c f x + f\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\left (-c f x +f \right )^{\frac {3}{2}}}\, dx \]
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 \[ -\frac {1}{2} \, {\left (\frac {c^{2} d^{3} x^{3}}{\sqrt {-c^{2} d f x^{2} + d f} f} + \frac {8 \, c d^{3} x^{2}}{\sqrt {-c^{2} d f x^{2} + d f} f} - \frac {17 \, d^{3} x}{\sqrt {-c^{2} d f x^{2} + d f} f} + \frac {15 \, d^{3} \arcsin \left (c x\right )}{\sqrt {d f} c f} - \frac {24 \, d^{3}}{\sqrt {-c^{2} d f x^{2} + d f} c f}\right )} a - \frac {\frac {{\left (c^{2} \int \frac {\sqrt {c x + 1} x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{\sqrt {-c x + 1} c x - \sqrt {-c x + 1}}\,{d x} + 2 \, c \int \frac {\sqrt {c x + 1} x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{\sqrt {-c x + 1} c x - \sqrt {-c x + 1}}\,{d x} + \int \frac {\sqrt {c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{{\left (c x - 1\right )} \sqrt {-c x + 1}}\,{d x}\right )} b d^{\frac {5}{2}}}{f}}{\sqrt {f}} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}}{{\left (f-c\,f\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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