Optimal. Leaf size=445 \[ \frac {4 \sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {4 \sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {32 \sqrt {(c+d x)^2+1} (c+d x)}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {8 c \sqrt {(c+d x)^2+1}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac {2 c \sqrt {(c+d x)^2+1}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 1.05, antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5865, 5803, 5655, 5774, 5779, 3308, 2180, 2204, 2205, 5667, 5665, 3307, 5675} \[ \frac {4 \sqrt {\pi } c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {4 \sqrt {\pi } c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {32 \sqrt {(c+d x)^2+1} (c+d x)}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {(c+d x)^2+1}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac {2 c \sqrt {(c+d x)^2+1}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3308
Rule 5655
Rule 5665
Rule 5667
Rule 5675
Rule 5774
Rule 5779
Rule 5803
Rule 5865
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d \left (a+b \sinh ^{-1}(x)\right )^{7/2}}+\frac {x}{d \left (a+b \sinh ^{-1}(x)\right )^{7/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}+\frac {4 \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 \operatorname {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}-\frac {(4 c) \operatorname {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {32 (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {32 \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {(8 c) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {32 (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {16 \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}+\frac {16 \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {(8 c) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {32 (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {32 \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}+\frac {32 \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}+\frac {(4 c) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}-\frac {(4 c) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{15 b^3 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {32 (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {8 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {(8 c) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}-\frac {(8 c) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{15 b^4 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{5 b d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}-\frac {4}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 c (c+d x)}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac {8 (c+d x)^2}{15 b^2 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 c \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {32 (c+d x) \sqrt {1+(c+d x)^2}}{15 b^3 d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {4 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {4 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}+\frac {8 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}\\ \end {align*}
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Mathematica [A] time = 2.59, size = 429, normalized size = 0.96 \[ \frac {\frac {\sqrt {b} \left (-\sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (16 \left (a+b \sinh ^{-1}(c+d x)\right )^2+3 b^2\right )-4 b \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )\right )}{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}+8 \sqrt {2 \pi } \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \sqrt {2 \pi } \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d^2}-\frac {c \left (-2 e^{-\sinh ^{-1}(c+d x)} \left (4 a^2+2 a b \left (4 \sinh ^{-1}(c+d x)-1\right )+b^2 \left (4 \sinh ^{-1}(c+d x)^2-2 \sinh ^{-1}(c+d x)+3\right )\right )+8 e^{a/b} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right )^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )-4 e^{-\frac {a}{b}} \left (a+b \sinh ^{-1}(c+d x)\right ) \left (e^{\frac {a}{b}+\sinh ^{-1}(c+d x)} \left (2 a+2 b \sinh ^{-1}(c+d x)+b\right )+2 b \left (-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )\right )-6 b^2 e^{\sinh ^{-1}(c+d x)}\right )}{30 b^3 d^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \]
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 \[ \int \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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