Optimal. Leaf size=328 \[ -\frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}-\frac {\sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {\sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
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Rubi [A] time = 1.24, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5866, 12, 5668, 5775, 5670, 5448, 3308, 2180, 2204, 2205, 5658} \[ -\frac {\sqrt {\pi } e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}-\frac {\sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {\sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5658
Rule 5668
Rule 5670
Rule 5775
Rule 5866
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}+\frac {\left (12 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (12 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (12 e^2\right ) \operatorname {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}-\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {4 e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {4 e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {4 e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}-\frac {e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 b^{5/2} d}+\frac {e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{2 b^{5/2} d}\\ \end {align*}
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Mathematica [A] time = 3.04, size = 391, normalized size = 1.19 \[ \frac {e^2 e^{-3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \left (-6 \sqrt {3} b e^{3 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-2 b e^{\frac {2 a}{b}+3 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )+2 e^{\frac {4 a}{b}+3 \cosh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+e^{\frac {3 a}{b}} \left (6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )-\left (e^{2 \cosh ^{-1}(c+d x)}+1\right ) \left (a \left (-4 e^{2 \cosh ^{-1}(c+d x)}+6 e^{4 \cosh ^{-1}(c+d x)}+6\right )+b \left (-4 e^{2 \cosh ^{-1}(c+d x)} \cosh ^{-1}(c+d x)+6 \cosh ^{-1}(c+d x)+e^{4 \cosh ^{-1}(c+d x)} \left (6 \cosh ^{-1}(c+d x)+1\right )-1\right )\right )\right )\right )}{12 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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 \[ e^{2} \left (\int \frac {c^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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