Optimal. Leaf size=337 \[ \frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3} \]
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Rubi [A]
time = 1.23, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5884, 5939,
5915, 5879, 5953, 3388, 2211, 2236, 2235, 3393} \begin {gather*} -\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {5 b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5879
Rule 5884
Rule 5915
Rule 5939
Rule 5953
Rubi steps
\begin {align*} \int x^2 \left (a+b \cosh ^{-1}(c x)\right )^{5/2} \, dx &=\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {1}{6} (5 b c) \int \frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac {1}{12} \left (5 b^2\right ) \int x^2 \sqrt {a+b \cosh ^{-1}(c x)} \, dx-\frac {(5 b) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c}\\ &=\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}+\frac {\left (5 b^2\right ) \int \sqrt {a+b \cosh ^{-1}(c x)} \, dx}{6 c^2}-\frac {1}{72} \left (5 b^3 c\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\cosh ^3(x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{72 c^3}-\frac {\left (5 b^3\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x} \sqrt {a+b \cosh ^{-1}(c x)}} \, dx}{12 c}\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \left (\frac {3 \cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{72 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{12 c^3}\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{288 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{96 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{24 c^3}\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{12 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{576 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{576 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{192 c^3}-\frac {\left (5 b^3\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c x)\right )}{192 c^3}\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {5 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {5 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{24 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{288 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{96 c^3}-\frac {\left (5 b^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c x)}\right )}{96 c^3}\\ &=\frac {5 b^2 x \sqrt {a+b \cosh ^{-1}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \cosh ^{-1}(c x)}-\frac {5 b \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{3/2}}{18 c}+\frac {1}{3} x^3 \left (a+b \cosh ^{-1}(c x)\right )^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right )}{576 c^3}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(924\) vs. \(2(337)=674\).
time = 9.23, size = 924, normalized size = 2.74 \begin {gather*} \frac {a^2 e^{-\frac {3 a}{b}} \sqrt {a+b \cosh ^{-1}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt {-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}}}+\frac {a \sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {a+b \cosh ^{-1}(c x)}+8 \sqrt {b} c x \cosh ^{-1}(c x) \sqrt {a+b \cosh ^{-1}(c x)}+(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \cosh ^{-1}(c x)} \left (2 \cosh ^{-1}(c x) \cosh \left (3 \cosh ^{-1}(c x)\right )-\sinh \left (3 \cosh ^{-1}(c x)\right )\right )\right )}{144 c^3}-\frac {27 \left (-4 b \sqrt {a+b \cosh ^{-1}(c x)} \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (a-5 b \cosh ^{-1}(c x)\right )+b c x \left (15+4 \cosh ^{-1}(c x)^2\right )\right )+\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+\sqrt {b} \left (12 a^2+12 a b+5 b^2\right ) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+\sqrt {b} \left (12 a^2-12 a b+5 b^2\right ) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )-12 b \sqrt {a+b \cosh ^{-1}(c x)} \left (b \left (5+12 \cosh ^{-1}(c x)^2\right ) \cosh \left (3 \cosh ^{-1}(c x)\right )+2 \left (a-5 b \cosh ^{-1}(c x)\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )\right )}{1728 c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{\frac {5}{2}}\, dx\]
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(-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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \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: RuntimeError} \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 x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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