Optimal. Leaf size=420 \[ -\frac {b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^5 \left (1-c^2 x^2\right )^{5/2}}{3 c (1+c x) (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {5 b f^5 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {28 b f^5 \left (1-c^2 x^2\right )^{5/2} \log (1+c x)}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}} \]
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Rubi [A]
time = 0.28, antiderivative size = 420, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {4763, 683,
655, 222, 4845, 641, 45, 4737} \begin {gather*} \frac {5 f^5 \left (1-c^2 x^2\right )^3 (a+b \text {ArcSin}(c x))}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}+\frac {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{3 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \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 {5 f^5 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x) (a+b \text {ArcSin}(c x))}{c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {5 b f^5 \left (1-c^2 x^2\right )^{5/2} \text {ArcSin}(c x)^2}{2 c (c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(c d x+d)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^5 \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 {28 b f^5 \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 45
Rule 222
Rule 641
Rule 655
Rule 683
Rule 4737
Rule 4763
Rule 4845
Rubi steps
\begin {align*} \int \frac {(f-c f x)^{5/2} \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)^5 \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 {2 f^5 (1-c x)^4 \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 {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{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 \left (\frac {5 f^5}{c}-\frac {2 f^5 (1-c x)^4}{3 c \left (1-c^2 x^2\right )^2}+\frac {10 f^5 (1-c x)^2}{3 c \left (1-c^2 x^2\right )}+\frac {5 f^5 \sin ^{-1}(c x)}{c \sqrt {1-c^2 x^2}}\right ) \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {5 b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \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 {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(1-c x)^4}{\left (1-c^2 x^2\right )^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (10 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(1-c x)^2}{1-c^2 x^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (5 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {\sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{(d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {5 b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {5 b f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \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 {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {(1-c x)^2}{(1+c x)^2} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (10 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \frac {1-c x}{1+c x} \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {5 b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {5 b f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \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 {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {\left (2 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (1+\frac {4}{(1+c x)^2}-\frac {4}{1+c x}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {\left (10 b f^5 \left (1-c^2 x^2\right )^{5/2}\right ) \int \left (-1+\frac {2}{1+c x}\right ) \, dx}{3 (d+c d x)^{5/2} (f-c f x)^{5/2}}\\ &=-\frac {b f^5 x \left (1-c^2 x^2\right )^{5/2}}{(d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {8 b f^5 \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 {5 b f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x)^2}{2 c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {2 f^5 (1-c x)^4 \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 {10 f^5 (1-c x)^2 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{3 c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}+\frac {5 f^5 \left (1-c^2 x^2\right )^{5/2} \sin ^{-1}(c x) \left (a+b \sin ^{-1}(c x)\right )}{c (d+c d x)^{5/2} (f-c f x)^{5/2}}-\frac {28 b f^5 \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 [B] Leaf count is larger than twice the leaf count of optimal. \(847\) vs. \(2(420)=840\).
time = 3.83, size = 847, normalized size = 2.02 \begin {gather*} \frac {f^2 \left (\frac {4 a \sqrt {d+c d x} \sqrt {f-c f x} \left (23+34 c x+3 c^2 x^2\right )}{(1+c x)^2}-60 a \sqrt {d} \sqrt {f} \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 )+\frac {2 b \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (-8+6 \text {ArcSin}(c x)+9 \text {ArcSin}(c x)^2-84 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+\cos \left (\frac {3}{2} \text {ArcSin}(c x)\right ) \left ((14-3 \text {ArcSin}(c x)) \text {ArcSin}(c x)+28 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+2 \left (-4+2 \left (2+7 \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)+3 \left (2+\sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-28 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{(1-c x) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}+\frac {2 b \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (\cos \left (\frac {3}{2} \text {ArcSin}(c x)\right ) \left (\text {ArcSin}(c x)+2 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )-\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right ) \left (4+3 \text {ArcSin}(c x)+6 \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right )+2 \left (-2+\left (2+\sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)-2 \left (2+\sqrt {1-c^2 x^2}\right ) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )}{(1-c x) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}+\frac {b \sqrt {d+c d x} \sqrt {f-c f x} \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right ) \left (2 \left (4+6 c x+6 c^2 x^2+52 (1+c x) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-18 \text {ArcSin}(c x)^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3+\text {ArcSin}(c x) \left (-24 \cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-35 \cos \left (\frac {3}{2} \text {ArcSin}(c x)\right )+3 \cos \left (\frac {5}{2} \text {ArcSin}(c x)\right )+24 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )-35 \sin \left (\frac {3}{2} \text {ArcSin}(c x)\right )-3 \sin \left (\frac {5}{2} \text {ArcSin}(c x)\right )\right )\right )}{(-1+c x) \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^4}\right )}{12 c d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.30, size = 0, normalized size = 0.00 \[\int \frac {\left (-c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )}{\left (c d x +d \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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(-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.
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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 )}^{5/2}}{{\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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