Optimal. Leaf size=315 \[ -\frac {25 b c x^2 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b c^3 x^4 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}} \]
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Rubi [A]
time = 0.19, antiderivative size = 315, 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, 4743,
4741, 4737, 30, 14, 267} \begin {gather*} \frac {5 x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))}{24 \left (1-c^2 x^2\right )}+\frac {5 x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))}{16 \left (1-c^2 x^2\right )^2}+\frac {5 (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))^2}{32 b c \left (1-c^2 x^2\right )^{5/2}}+\frac {1}{6} x (c d x+d)^{5/2} (f-c f x)^{5/2} (a+b \text {ArcSin}(c x))-\frac {25 b c x^2 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{5/2} (f-c f x)^{5/2}}{36 c}+\frac {5 b c^3 x^4 (c d x+d)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 267
Rule 4737
Rule 4741
Rule 4743
Rule 4763
Rubi steps
\begin {align*} \int (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {\left ((d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \left (1-c^2 x^2\right )^2 \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \left (1-c^2 x^2\right ) \, dx}{24 \left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int \left (x-c^2 x^3\right ) \, dx}{24 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (f-c f x)^{5/2}\right ) \int x \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {25 b c x^2 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b c^3 x^4 (d+c d x)^{5/2} (f-c f x)^{5/2}}{96 \left (1-c^2 x^2\right )^{5/2}}+\frac {b (d+c d x)^{5/2} (f-c f x)^{5/2} \sqrt {1-c^2 x^2}}{36 c}+\frac {1}{6} x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (f-c f x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{32 b c \left (1-c^2 x^2\right )^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.98, size = 303, normalized size = 0.96 \begin {gather*} \frac {d^2 f^2 \left (360 b \sqrt {d+c d x} \sqrt {f-c f x} \text {ArcSin}(c x)^2-720 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 )+\sqrt {d+c d x} \sqrt {f-c f x} \left (1584 a c x \sqrt {1-c^2 x^2}-1248 a c^3 x^3 \sqrt {1-c^2 x^2}+384 a c^5 x^5 \sqrt {1-c^2 x^2}+270 b \cos (2 \text {ArcSin}(c x))+27 b \cos (4 \text {ArcSin}(c x))+2 b \cos (6 \text {ArcSin}(c x))\right )+12 b \sqrt {d+c d x} \sqrt {f-c f x} \text {ArcSin}(c x) (45 \sin (2 \text {ArcSin}(c x))+9 \sin (4 \text {ArcSin}(c x))+\sin (6 \text {ArcSin}(c x)))\right )}{2304 c \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 \[\int \left (c d x +d \right )^{\frac {5}{2}} \left (-c f x +f \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\right )}^{5/2}\,{\left (f-c\,f\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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