Optimal. Leaf size=109 \[ -\frac {2 (a+b \text {ArcCos}(c x))^2}{3 d (d x)^{3/2}}+\frac {8 b c (a+b \text {ArcCos}(c x)) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right )}{3 d^2 \sqrt {d x}}+\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4724, 4806}
\begin {gather*} \frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}+\frac {8 b c \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right ) (a+b \text {ArcCos}(c x))}{3 d^2 \sqrt {d x}}-\frac {2 (a+b \text {ArcCos}(c x))^2}{3 d (d x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 4724
Rule 4806
Rubi steps
\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}-\frac {(4 b c) \int \frac {a+b \cos ^{-1}(c x)}{(d x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 d}\\ &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^2}{3 d (d x)^{3/2}}+\frac {8 b c \left (a+b \cos ^{-1}(c x)\right ) \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right )}{3 d^2 \sqrt {d x}}+\frac {16 b^2 c^2 \sqrt {d x} \, _3F_2\left (\frac {1}{4},\frac {1}{4},1;\frac {3}{4},\frac {5}{4};c^2 x^2\right )}{3 d^3}\\ \end {align*}
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Mathematica [A]
time = 10.55, size = 198, normalized size = 1.82 \begin {gather*} \frac {x \left (-8 \text {Gamma}\left (\frac {7}{4}\right ) \text {Gamma}\left (\frac {9}{4}\right ) \left (3 \left (a^2-8 b^2 c^2 x^2+2 b \left (a-2 b c x \sqrt {1-c^2 x^2}\right ) \text {ArcCos}(c x)+b^2 \text {ArcCos}(c x)^2\right )-12 a b c x \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};c^2 x^2\right )-4 b^2 c^3 x^3 \sqrt {1-c^2 x^2} \text {ArcCos}(c x) \, _2F_1\left (1,\frac {5}{4};\frac {7}{4};c^2 x^2\right )\right )+3 \sqrt {2} b^2 c^4 \pi x^4 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};c^2 x^2\right )\right )}{36 (d x)^{5/2} \text {Gamma}\left (\frac {7}{4}\right ) \text {Gamma}\left (\frac {9}{4}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{2}}{\left (d x \right )^{\frac {5}{2}}}\, dx\]
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: TypeError} \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 {acos}\left (c\,x\right )\right )}^2}{{\left (d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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