Optimal. Leaf size=648 \[ -\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b c \left (7 c^2 d^2+3 e^2\right ) \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 \sqrt {-c^2} e}{-c^2 d+\sqrt {-c^2} e}\right )}{15 \sqrt {-c^2} d^2 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}-\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} F\left (\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \Pi \left (2;\text {ArcSin}\left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|\frac {2 e}{\sqrt {-c^2} d+e}\right )}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
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Rubi [A]
time = 0.73, antiderivative size = 785, normalized size of antiderivative = 1.21, number of steps
used = 19, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {6425, 1588,
972, 759, 849, 858, 733, 435, 430, 21, 947, 174, 552, 551} \begin {gather*} -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \sqrt {-c^2} \sqrt {c^2 x^2+1} \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}} F\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 )}{15 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}+\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 )}{5 c d^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}}+\frac {16 b c \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 )}{15 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt {\frac {d+e x}{\frac {e}{\sqrt {-c^2}}+d}}}+\frac {4 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 )}{5 c d^2 e x \sqrt {\frac {1}{c^2 x^2}+1} \sqrt {d+e x}}-\frac {4 b e \left (c^2 x^2+1\right )}{5 c d^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {16 b c e \left (c^2 x^2+1\right )}{15 x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right )^2 \sqrt {d+e x}}-\frac {4 b e \left (c^2 x^2+1\right )}{15 c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 d^2+e^2\right ) (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 174
Rule 430
Rule 435
Rule 551
Rule 552
Rule 733
Rule 759
Rule 849
Rule 858
Rule 947
Rule 972
Rule 1588
Rule 6425
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {(2 b) \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{5/2}} \, dx}{5 c e}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x (d+e x)^{5/2} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \left (-\frac {e}{d (d+e x)^{5/2} \sqrt {\frac {1}{c^2}+x^2}}-\frac {e}{d^2 (d+e x)^{3/2} \sqrt {\frac {1}{c^2}+x^2}}+\frac {1}{d^2 x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}}\right ) \, dx}{5 c e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{(d+e x)^{5/2} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 c d \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{x \sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (4 b c \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {d}{2}-\frac {e x}{2}}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (4 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {-\frac {3 d}{2}+\frac {e x}{2}}{(d+e x)^{3/2} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2+\frac {e^2}{c^2}\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {\left (2 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}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (2 b c \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {\frac {1}{c^2}+x^2}} \, dx}{5 d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b c \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\frac {1}{4} \left (3 d^2-\frac {e^2}{c^2}\right )+d e x}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{15 d \left (c^2 d^2+e^2\right ) \left (d^2+\frac {e^2}{c^2}\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (4 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 )}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x}\\ &=-\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (2 b \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {\frac {1}{c^2}+x^2}} \, dx}{15 c d \left (d^2+\frac {e^2}{c^2}\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}+\frac {\left (8 b c \sqrt {\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {\frac {1}{c^2}+x^2}} \, dx}{15 \left (c^2 d^2+e^2\right ) \left (d^2+\frac {e^2}{c^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 \left (d-\frac {\sqrt {-c^2} e}{c^2}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d-\frac {\sqrt {-c^2} e}{c^2}}}}+\frac {\left (4 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 )}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\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 )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}+\frac {4 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 )}{5 c d^2 e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {\left (16 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 \left (d-\frac {\sqrt {-c^2} e}{c^2}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c \left (c^2 d^2+e^2\right ) \left (d^2+\frac {e^2}{c^2}\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d-\frac {\sqrt {-c^2} e}{c^2}}}}-\frac {\left (4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d-\frac {\sqrt {-c^2} e}{c^2}}} \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {-c^2} e x^2}{c^2 \left (d-\frac {\sqrt {-c^2} e}{c^2}\right )}}} \, dx,x,\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )}{15 c^3 d \left (d^2+\frac {e^2}{c^2}\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}\\ &=-\frac {4 b e \left (1+c^2 x^2\right )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x (d+e x)^{3/2}}-\frac {16 b c e \left (1+c^2 x^2\right )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {4 b e \left (1+c^2 x^2\right )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {16 b c \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 )}{15 \left (c^2 d^2+e^2\right )^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}+\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 )}{5 c d^2 \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}}}-\frac {4 b \sqrt {-c^2} \sqrt {\frac {d+e x}{d+\frac {e}{\sqrt {-c^2}}}} \sqrt {1+c^2 x^2} F\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 )}{15 c d \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 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 )}{5 c d^2 e \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 = 30.05, size = 1217, normalized size = 1.88 \begin {gather*} -\frac {2 a}{5 e (d+e x)^{5/2}}+\frac {b \left (-\frac {c^4 \left (e+\frac {d}{x}\right )^4 x^4 \left (-\frac {4 \left (7 c^2 d^2+3 e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}}}{15 c^2 d^2 \left (c^2 d^2+e^2\right )^2}+\frac {2 \text {csch}^{-1}(c x)}{5 c^3 d^3 e}-\frac {2 e^2 \text {csch}^{-1}(c x)}{5 c^3 d^3 \left (e+\frac {d}{x}\right )^3}+\frac {2 \left (-2 c d e^2 \sqrt {1+\frac {1}{c^2 x^2}}+9 c^2 d^2 e \text {csch}^{-1}(c x)+9 e^3 \text {csch}^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2+e^2\right ) \left (e+\frac {d}{x}\right )^2}-\frac {2 \left (-16 c^3 d^3 e \sqrt {1+\frac {1}{c^2 x^2}}-8 c d e^3 \sqrt {1+\frac {1}{c^2 x^2}}+9 c^4 d^4 \text {csch}^{-1}(c x)+18 c^2 d^2 e^2 \text {csch}^{-1}(c x)+9 e^4 \text {csch}^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2+e^2\right )^2 \left (e+\frac {d}{x}\right )}\right )}{(d+e x)^{7/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{7/2} (c x)^{7/2} \left (-\frac {\sqrt {2} \left (c^2 d^2 e+e^3\right ) \sqrt {1+i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{\sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2} \sqrt {\frac {e (1-i c x)}{i c d+e}}}+\frac {i \sqrt {2} (c d-i e) \left (3 c^3 d^3-c d e^2\right ) \sqrt {1+i c x} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )}{e \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {2 \left (-7 c^2 d^2 e-3 e^3\right ) \cosh \left (2 \text {csch}^{-1}(c x)\right ) \left (-\left ((c d+c e x) \left (1+c^2 x^2\right )\right )+\frac {c x \left (c d \sqrt {2+2 i c x} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} F\left (\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )+2 \sqrt {-\frac {e (-i+c x)}{c d+i e}} (i+c x) \sqrt {\frac {c d+c e x}{c d-i e}} \left ((c d+i e) E\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i e F\left (\text {ArcSin}\left (\sqrt {\frac {c d+c e x}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )\right )+(i c d+e) \sqrt {2+2 i c x} \sqrt {-\frac {e (i+c x)}{c d-i e}} \sqrt {\frac {e (i+c x) (c d+c e x)}{(i c d+e)^2}} \Pi \left (1+\frac {i c d}{e};\text {ArcSin}\left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right )|\frac {i c d+e}{2 e}\right )\right )}{2 \sqrt {-\frac {e (i+c x)}{c d-i e}}}\right )}{c d \sqrt {1+\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (2+c^2 x^2\right )}\right )}{15 c d e \left (c^2 d^2+e^2\right )^2 (d+e x)^{7/2}}\right )}{c} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 0.98, size = 3782, normalized size = 5.84
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(3782\) |
default | \(\text {Expression too large to display}\) | \(3782\) |
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(-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 \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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