Optimal. Leaf size=125 \[ -\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}-\frac {4 b c^{3/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}+\frac {4 b c^{3/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}+\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}} \]
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Rubi [A] time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4628, 325, 329, 307, 221, 1199, 424} \[ -\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {4 b c^{3/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}+\frac {4 b c^{3/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}} \]
Antiderivative was successfully verified.
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Rule 221
Rule 307
Rule 325
Rule 329
Rule 424
Rule 1199
Rule 4628
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{(d x)^{5/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}-\frac {(2 b c) \int \frac {1}{(d x)^{3/2} \sqrt {1-c^2 x^2}} \, dx}{3 d}\\ &=\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {\left (2 b c^3\right ) \int \frac {\sqrt {d x}}{\sqrt {1-c^2 x^2}} \, dx}{3 d^3}\\ &=\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {\left (4 b c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{3 d^4}\\ &=\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}-\frac {\left (4 b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{3 d^3}+\frac {\left (4 b c^2\right ) \operatorname {Subst}\left (\int \frac {1+\frac {c x^2}{d}}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}-\frac {4 b c^{3/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}+\frac {\left (4 b c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {c x^2}{d}}}{\sqrt {1-\frac {c x^2}{d}}} \, dx,x,\sqrt {d x}\right )}{3 d^3}\\ &=\frac {4 b c \sqrt {1-c^2 x^2}}{3 d^2 \sqrt {d x}}-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{3 d (d x)^{3/2}}+\frac {4 b c^{3/2} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}-\frac {4 b c^{3/2} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{3 d^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 68, normalized size = 0.54 \[ \frac {2 x \left (2 b c^3 x^3 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};c^2 x^2\right )-3 \left (a-2 b c x \sqrt {1-c^2 x^2}+b \cos ^{-1}(c x)\right )\right )}{9 (d x)^{5/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {d x} {\left (b \arccos \left (c x\right ) + a\right )}}{d^{3} x^{3}}, x\right ) \]
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 \[ \int \frac {b \arccos \left (c x\right ) + a}{\left (d x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 129, normalized size = 1.03 \[ \frac {-\frac {2 a}{3 \left (d x \right )^{\frac {3}{2}}}+2 b \left (-\frac {\arccos \left (c x \right )}{3 \left (d x \right )^{\frac {3}{2}}}-\frac {2 c \left (-\frac {\sqrt {-c^{2} x^{2}+1}}{\sqrt {d x}}+\frac {c \sqrt {-c x +1}\, \sqrt {c x +1}\, \left (\EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )-\EllipticE \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{3 d}\right )}{d} \]
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 \[ -\frac {2 \, b \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - {\left (2 \, b c d^{3} x \int \frac {\sqrt {c x + 1} \sqrt {-c x + 1} \sqrt {x}}{c^{2} d^{3} x^{4} - d^{3} x^{2}}\,{d x} + {\left (2 \, b c x \arctan \left (\frac {1}{\sqrt {c} \sqrt {x}}\right ) + b c x \log \left (-\frac {c x - 1}{c x + 2 \, \sqrt {c} \sqrt {x} + 1}\right )\right )} \sqrt {c}\right )} \sqrt {x}}{3 \, d^{\frac {5}{2}} x^{\frac {3}{2}}} \]
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 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d\,x\right )}^{5/2}} \,d x \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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