Optimal. Leaf size=146 \[ -\frac {b c \sqrt {1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}} \]
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Rubi [A]
time = 0.12, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {198, 197, 5792,
12, 585, 79, 65, 223, 212} \begin {gather*} \frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2+1}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}-\frac {b c \sqrt {c^2 x^2+1}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 212
Rule 223
Rule 585
Rule 5792
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+e x^2\right )^{5/2}} \, dx &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-(b c) \int \frac {x \left (3 d+2 e x^2\right )}{3 d^2 \sqrt {1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \int \frac {x \left (3 d+2 e x^2\right )}{\sqrt {1+c^2 x^2} \left (d+e x^2\right )^{3/2}} \, dx}{3 d^2}\\ &=\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {3 d+2 e x}{\sqrt {1+c^2 x} (d+e x)^{3/2}} \, dx,x,x^2\right )}{6 d^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {1}{\sqrt {1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{3 d^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {e}{c^2}+\frac {e x^2}{c^2}}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{3 c d^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {(2 b) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{c^2}} \, dx,x,\frac {\sqrt {1+c^2 x^2}}{\sqrt {d+e x^2}}\right )}{3 c d^2}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{3 d \left (c^2 d-e\right ) \sqrt {d+e x^2}}+\frac {x \left (a+b \sinh ^{-1}(c x)\right )}{3 d \left (d+e x^2\right )^{3/2}}+\frac {2 x \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+e x^2}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{3 d^2 \sqrt {e}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.22, size = 139, normalized size = 0.95 \begin {gather*} \frac {-\frac {b c d \sqrt {1+c^2 x^2} \left (d+e x^2\right )}{c^2 d-e}+a x \left (3 d+2 e x^2\right )-b c x^2 \left (d+e x^2\right ) \sqrt {1+\frac {e x^2}{d}} F_1\left (1;\frac {1}{2},\frac {1}{2};2;-c^2 x^2,-\frac {e x^2}{d}\right )+b x \left (3 d+2 e x^2\right ) \sinh ^{-1}(c x)}{3 d^2 \left (d+e x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {a +b \arcsinh \left (c x \right )}{\left (e \,x^{2}+d \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 970 vs.
\(2 (125) = 250\).
time = 0.43, size = 970, normalized size = 6.64 \begin {gather*} \frac {{\left (b x^{4} \cosh \left (1\right )^{3} + b x^{4} \sinh \left (1\right )^{3} - b c^{2} d^{3} - {\left (b c^{2} d x^{4} - 2 \, b d x^{2}\right )} \cosh \left (1\right )^{2} - {\left (b c^{2} d x^{4} - 3 \, b x^{4} \cosh \left (1\right ) - 2 \, b d x^{2}\right )} \sinh \left (1\right )^{2} - {\left (2 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} \cosh \left (1\right ) - {\left (2 \, b c^{2} d^{2} x^{2} - 3 \, b x^{4} \cosh \left (1\right )^{2} - b d^{2} + 2 \, {\left (b c^{2} d x^{4} - 2 \, b d x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} \log \left (c^{4} d^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )^{2} + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \sinh \left (1\right )^{2} - 4 \, {\left (c^{3} d + {\left (2 \, c^{3} x^{2} + c\right )} \cosh \left (1\right ) + {\left (2 \, c^{3} x^{2} + c\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \sqrt {\cosh \left (1\right ) + \sinh \left (1\right )} + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d\right )} \cosh \left (1\right ) + 2 \, {\left (4 \, c^{4} d x^{2} + 3 \, c^{2} d + {\left (8 \, c^{4} x^{4} + 8 \, c^{2} x^{2} + 1\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right ) - 2 \, {\left (3 \, b c^{2} d^{2} x \cosh \left (1\right ) - 2 \, b x^{3} \cosh \left (1\right )^{3} - 2 \, b x^{3} \sinh \left (1\right )^{3} + {\left (2 \, b c^{2} d x^{3} - 3 \, b d x\right )} \cosh \left (1\right )^{2} + {\left (2 \, b c^{2} d x^{3} - 6 \, b x^{3} \cosh \left (1\right ) - 3 \, b d x\right )} \sinh \left (1\right )^{2} + {\left (3 \, b c^{2} d^{2} x - 6 \, b x^{3} \cosh \left (1\right )^{2} + 2 \, {\left (2 \, b c^{2} d x^{3} - 3 \, b d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (3 \, a c^{2} d^{2} x \cosh \left (1\right ) - 2 \, a x^{3} \cosh \left (1\right )^{3} - 2 \, a x^{3} \sinh \left (1\right )^{3} + {\left (2 \, a c^{2} d x^{3} - 3 \, a d x\right )} \cosh \left (1\right )^{2} + {\left (2 \, a c^{2} d x^{3} - 6 \, a x^{3} \cosh \left (1\right ) - 3 \, a d x\right )} \sinh \left (1\right )^{2} + {\left (3 \, a c^{2} d^{2} x - 6 \, a x^{3} \cosh \left (1\right )^{2} + 2 \, {\left (2 \, a c^{2} d x^{3} - 3 \, a d x\right )} \cosh \left (1\right )\right )} \sinh \left (1\right ) - {\left (b c d x^{2} \cosh \left (1\right )^{2} + b c d x^{2} \sinh \left (1\right )^{2} + b c d^{2} \cosh \left (1\right ) + {\left (2 \, b c d x^{2} \cosh \left (1\right ) + b c d^{2}\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {x^{2} \cosh \left (1\right ) + x^{2} \sinh \left (1\right ) + d}}{6 \, {\left (d^{2} x^{4} \cosh \left (1\right )^{4} + d^{2} x^{4} \sinh \left (1\right )^{4} - c^{2} d^{5} \cosh \left (1\right ) - {\left (c^{2} d^{3} x^{4} - 2 \, d^{3} x^{2}\right )} \cosh \left (1\right )^{3} - {\left (c^{2} d^{3} x^{4} - 4 \, d^{2} x^{4} \cosh \left (1\right ) - 2 \, d^{3} x^{2}\right )} \sinh \left (1\right )^{3} - {\left (2 \, c^{2} d^{4} x^{2} - d^{4}\right )} \cosh \left (1\right )^{2} - {\left (2 \, c^{2} d^{4} x^{2} - 6 \, d^{2} x^{4} \cosh \left (1\right )^{2} - d^{4} + 3 \, {\left (c^{2} d^{3} x^{4} - 2 \, d^{3} x^{2}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + {\left (4 \, d^{2} x^{4} \cosh \left (1\right )^{3} - c^{2} d^{5} - 3 \, {\left (c^{2} d^{3} x^{4} - 2 \, d^{3} x^{2}\right )} \cosh \left (1\right )^{2} - 2 \, {\left (2 \, c^{2} d^{4} x^{2} - d^{4}\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \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 = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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