Optimal. Leaf size=393 \[ \frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Rubi [A]
time = 1.48, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {45, 6445, 12,
6853, 6874, 759, 21, 733, 435, 972, 946, 174, 552, 551} \begin {gather*} -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {d+e x} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {c^2 x^2+1} \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}+\frac {4 b \left (c^2 x^2+1\right )}{3 c x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 21
Rule 45
Rule 174
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 946
Rule 972
Rule 6445
Rule 6853
Rule 6874
Rubi steps
\begin {align*} \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {b \int \frac {2 (-2 d-3 e x)}{3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{c}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {(2 b) \int \frac {-2 d-3 e x}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}} \, dx}{3 c e^2}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {-2 d-3 e x}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {3 e}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}}-\frac {2 d}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x (d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b d \sqrt {1+c^2 x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{3/2} \sqrt {1+c^2 x^2}}+\frac {1}{d x \sqrt {d+e x} \sqrt {1+c^2 x^2}}\right ) \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {1+c^2 x^2}} \, dx}{3 c e \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x \sqrt {1-\sqrt {-c^2} x} \sqrt {1+\sqrt {-c^2} x} \sqrt {d+e x}} \, dx}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (8 b c \sqrt {1+c^2 x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {1+c^2 x^2}} \, dx}{3 e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{\sqrt {-c^2}}-\frac {e x^2}{\sqrt {-c^2}}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 b c \sqrt {1+c^2 x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1+c^2 x^2}} \, dx}{3 e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (8 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 d-\sqrt {-c^2} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {\left (8 b \sqrt {1+c^2 x^2} \sqrt {1+\frac {e \left (-1+\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{\sqrt {-c^2} \left (d+\frac {e}{\sqrt {-c^2}}\right )}}} \, dx,x,\sqrt {1-\sqrt {-c^2} x}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {1+c^2 x^2} \sqrt {1-\frac {e \left (1-\sqrt {-c^2} x\right )}{\sqrt {-c^2} d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 11.62, size = 390, normalized size = 0.99 \begin {gather*} \frac {2}{3} \left (\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} x}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {a (2 d+3 e x)}{e^2 (d+e x)^{3/2}}-\frac {b (2 d+3 e x) \text {csch}^{-1}(c x)}{e^2 (d+e x)^{3/2}}+\frac {2 i b \sqrt {-\frac {c}{c d-i e}} \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left (c d E\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )-c d F\left (i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+2 (c d-i e) \Pi \left (1-\frac {i e}{c d};i \sinh ^{-1}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )\right )}{c^2 d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.80, size = 2106, normalized size = 5.36
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2106\) |
default | \(\text {Expression too large to display}\) | \(2106\) |
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\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 = -0.00 \begin {gather*} \int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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