Optimal. Leaf size=394 \[ -\frac {15 \sqrt {\pi } b^{5/2} e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {15 \sqrt {\pi } b^{5/2} e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {5 b e^2 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {5 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d} \]
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Rubi [A] time = 1.24, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5865, 12, 5663, 5758, 5717, 5653, 5779, 3308, 2180, 2204, 2205, 3312} \[ -\frac {15 \sqrt {\pi } b^{5/2} e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {15 \sqrt {\pi } b^{5/2} e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 \sqrt {\frac {\pi }{3}} b^{5/2} e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {5 b e^2 \sqrt {(c+d x)^2+1} (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {5 b e^2 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 3312
Rule 5653
Rule 5663
Rule 5717
Rule 5758
Rule 5779
Rule 5865
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^2 x^2 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int x^2 \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b e^2\right ) \operatorname {Subst}\left (\int \frac {x^3 \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b e^2\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int x^2 \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{12 d}\\ &=\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{6 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh ^3(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 i b^3 e^2\right ) \operatorname {Subst}\left (\int \left (\frac {3 i \sinh (x)}{4 \sqrt {a+b x}}-\frac {i \sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{288 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{96 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{24 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{24 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{12 d}+\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{12 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{576 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{576 d}-\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{192 d}+\frac {\left (5 b^3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{192 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {5 b^{5/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}+\frac {5 b^{5/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{24 d}+\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{288 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{288 d}-\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{96 d}+\frac {\left (5 b^2 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{96 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sinh ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {15 b^{5/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {5 b^{5/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {15 b^{5/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{576 d}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 238, normalized size = 0.60 \[ -\frac {e^2 e^{-\frac {3 a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \left (81 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {7}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {7}{2},-\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-81 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {7}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {7}{2},\frac {3 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{648 d \left (-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}\right )^{3/2}} \]
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 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int \left (d e x +c e \right )^{2} \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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 \[ e^{2} \left (\int a^{2} c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b^{2} c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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