Optimal. Leaf size=509 \[ -\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{64 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.40, antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5786, 5785,
5783, 5776, 5812, 30, 5798, 14, 267} \begin {gather*} -\frac {865 a c^2 x^2 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt {a^2 c x^2+c}}{216 a}-\frac {15 a c^2 x^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 \sqrt {a^2 x^2+1}}+\frac {5}{16} c^2 x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^3+\frac {245}{384} c^2 x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)+\frac {1}{36} c^2 x \left (a^2 x^2+1\right )^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)+\frac {65}{576} c^2 x \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)+\frac {5 c^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^4}{64 a \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{12 a}-\frac {5 c^2 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{32 a}-\frac {115 c^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^2}{768 a \sqrt {a^2 x^2+1}}+\frac {1}{6} x \left (a^2 c x^2+c\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \sinh ^{-1}(a x)^3-\frac {65 a^3 c^2 x^4 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 267
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5812
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3 \, dx &=\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \sinh ^{-1}(a x)^2 \, dx}{2 \sqrt {1+a^2 x^2}}\\ &=-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3 \, dx+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{5/2} \sinh ^{-1}(a x) \, dx}{6 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \sinh ^{-1}(a x)^2 \, dx}{8 \sqrt {1+a^2 x^2}}\\ &=\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x) \, dx}{36 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \sinh ^{-1}(a x) \, dx}{16 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \, dx}{36 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \sinh ^{-1}(a x)^2 \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \, dx}{48 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \sinh ^{-1}(a x) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{64 \sqrt {1+a^2 x^2}}+\frac {\left (15 a^2 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{96 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sinh ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{96 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{32 \sqrt {1+a^2 x^2}}\\ &=-\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \sinh ^{-1}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \sinh ^{-1}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^4}{64 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 177, normalized size = 0.35 \begin {gather*} \frac {c^2 \sqrt {c+a^2 c x^2} \left (4320 \sinh ^{-1}(a x)^4-9720 \cosh \left (2 \sinh ^{-1}(a x)\right )-243 \cosh \left (4 \sinh ^{-1}(a x)\right )-8 \cosh \left (6 \sinh ^{-1}(a x)\right )-72 \sinh ^{-1}(a x)^2 \left (270 \cosh \left (2 \sinh ^{-1}(a x)\right )+27 \cosh \left (4 \sinh ^{-1}(a x)\right )+2 \cosh \left (6 \sinh ^{-1}(a x)\right )\right )+288 \sinh ^{-1}(a x)^3 \left (45 \sinh \left (2 \sinh ^{-1}(a x)\right )+9 \sinh \left (4 \sinh ^{-1}(a x)\right )+\sinh \left (6 \sinh ^{-1}(a x)\right )\right )+12 \sinh ^{-1}(a x) \left (1620 \sinh \left (2 \sinh ^{-1}(a x)\right )+81 \sinh \left (4 \sinh ^{-1}(a x)\right )+4 \sinh \left (6 \sinh ^{-1}(a x)\right )\right )\right )}{55296 a \sqrt {1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.37, size = 802, normalized size = 1.58
method | result | size |
default | \(\frac {5 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )^{4} c^{2}}{64 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}+32 \sqrt {a^{2} x^{2}+1}\, a^{6} x^{6}+64 a^{5} x^{5}+48 a^{4} \sqrt {a^{2} x^{2}+1}\, x^{4}+38 a^{3} x^{3}+18 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+6 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (36 \arcsinh \left (a x \right )^{3}-18 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )-1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}+8 a^{4} \sqrt {a^{2} x^{2}+1}\, x^{4}+12 a^{3} x^{3}+8 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+4 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (32 \arcsinh \left (a x \right )^{3}-24 \arcsinh \left (a x \right )^{2}+12 \arcsinh \left (a x \right )-3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \arcsinh \left (a x \right )^{3}-6 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )-3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \arcsinh \left (a x \right )^{3}+6 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )+3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} \sqrt {a^{2} x^{2}+1}\, x^{4}+12 a^{3} x^{3}-8 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+4 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (32 \arcsinh \left (a x \right )^{3}+24 \arcsinh \left (a x \right )^{2}+12 \arcsinh \left (a x \right )+3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}-32 \sqrt {a^{2} x^{2}+1}\, a^{6} x^{6}+64 a^{5} x^{5}-48 a^{4} \sqrt {a^{2} x^{2}+1}\, x^{4}+38 a^{3} x^{3}-18 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+6 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (36 \arcsinh \left (a x \right )^{3}+18 \arcsinh \left (a x \right )^{2}+6 \arcsinh \left (a x \right )+1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}\) | \(802\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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