Optimal. Leaf size=427 \[ -\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b d \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}-\frac {b e x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+\frac {b e \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 1.26, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5706, 5653, 5717, 5657, 3307, 2180, 2205, 2204, 5663, 5758, 5669, 5448} \[ -\frac {3 \sqrt {\pi } b^{3/2} e e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 \sqrt {\pi } b^{3/2} e e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {\sqrt {\frac {\pi }{3}} b^{3/2} e e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 \sqrt {\pi } b^{3/2} d e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 \sqrt {\pi } b^{3/2} d e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b d \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}-\frac {b e x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+\frac {b e \sqrt {c^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5448
Rule 5653
Rule 5657
Rule 5663
Rule 5669
Rule 5706
Rule 5717
Rule 5758
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac {1}{2} (3 b c d) \int \frac {x \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} (b c e) \int \frac {x^3 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{4} \left (3 b^2 d\right ) \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx+\frac {1}{12} \left (b^2 e\right ) \int \frac {x^2}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx+\frac {(b e) \int \frac {x \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {1+c^2 x^2}} \, dx}{3 c}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b d) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{4 c}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}-\frac {\left (b^2 e\right ) \int \frac {1}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{6 c^2}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b d) \operatorname {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac {(3 b d) \operatorname {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{8 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{6 c^3}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}+\frac {\cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {(3 b d) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c}+\frac {(3 b d) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{4 c}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac {(b e) \operatorname {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{12 c^3}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {(b e) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{6 c^3}-\frac {(b e) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{6 c^3}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}+\frac {\left (b^2 e\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{12 c^3}+\frac {(b e) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{48 c^3}-\frac {(b e) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{48 c^3}-\frac {(b e) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{48 c^3}+\frac {(b e) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{48 c^3}\\ &=-\frac {3 b d \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{2 c}+\frac {b e \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac {b e x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}+\frac {3 b^{3/2} d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{8 c}-\frac {3 b^{3/2} e e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{96 c^3}\\ \end {align*}
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Mathematica [A] time = 4.84, size = 770, normalized size = 1.80 \[ \frac {a e e^{-\frac {3 a}{b}} \sqrt {a+b \sinh ^{-1}(c x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt {-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}}+\frac {\sqrt {b} d \left (4 \sqrt {b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt {c^2 x^2+1}\right ) \sqrt {a+b \sinh ^{-1}(c x)}+\sqrt {\pi } (3 b-2 a) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )\right )}{8 c}+\frac {\sqrt {b} e \left (-9 \left (4 \sqrt {b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt {c^2 x^2+1}\right ) \sqrt {a+b \sinh ^{-1}(c x)}+\sqrt {\pi } (3 b-2 a) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )\right )+\sqrt {3 \pi } (b-2 a) \left (\sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {3 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+\sqrt {3 \pi } (2 a+b) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )+12 \sqrt {b} \left (2 \sinh ^{-1}(c x) \sinh \left (3 \sinh ^{-1}(c x)\right )-\cosh \left (3 \sinh ^{-1}(c x)\right )\right ) \sqrt {a+b \sinh ^{-1}(c x)}\right )}{288 c^3}+\frac {a d e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c x)} \left (\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}}}-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c x)}}\right )}{2 c} \]
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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \left (e \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}\,\left (e\,x^2+d\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}} \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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