Optimal. Leaf size=216 \[ -\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {105 b^{7/2} e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {105 b^{7/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d} \]
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Rubi [A]
time = 0.28, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5858, 5772,
5798, 5774, 3388, 2211, 2236, 2235} \begin {gather*} \frac {105 \sqrt {\pi } b^{7/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {105 \sqrt {\pi } b^{7/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d}-\frac {105 b^3 \sqrt {(c+d x)^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5772
Rule 5774
Rule 5798
Rule 5858
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {(7 b) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^{5/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (35 b^2\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}-\frac {\left (105 b^3\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \text {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+d x)\right )}{32 d}+\frac {\left (105 b^3\right ) \text {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+d x)\right )}{32 d}\\ &=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {\left (105 b^3\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{16 d}+\frac {\left (105 b^3\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{16 d}\\ &=-\frac {105 b^3 \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {35 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {7 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}}{d}+\frac {105 b^{7/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {105 b^{7/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{32 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(698\) vs. \(2(216)=432\).
time = 3.12, size = 698, normalized size = 3.23 \begin {gather*} \frac {16 a^3 e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}}}\right )+12 a^2 \sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (-3 \sqrt {1+(c+d x)^2}+2 (c+d x) \sinh ^{-1}(c+d x)\right )+(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d 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 \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+6 a \sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (2 \sqrt {1+(c+d x)^2} \left (a-5 b \sinh ^{-1}(c+d x)\right )+b (c+d x) \left (15+4 \sinh ^{-1}(c+d x)^2\right )\right )+\left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+\sqrt {b} \left (4 \sqrt {b} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (2 b (c+d x) \left (-10 a+35 b \sinh ^{-1}(c+d x)+4 b \sinh ^{-1}(c+d x)^3\right )+\sqrt {1+(c+d x)^2} \left (-4 a^2+4 a b \sinh ^{-1}(c+d x)-7 b^2 \left (15+4 \sinh ^{-1}(c+d x)^2\right )\right )\right )+\left (8 a^3+36 a^2 b+90 a b^2+105 b^3\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+\left (-8 a^3+36 a^2 b-90 a b^2+105 b^3\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {7}{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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