Optimal. Leaf size=540 \[ -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}}-\frac {4 b \sqrt {1-c^2 x^2} \left (7 c^2 d^2-3 e^2\right ) \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^3-d e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}} \]
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Rubi [A] time = 0.75, antiderivative size = 637, normalized size of antiderivative = 1.18, 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 = {5227, 1574, 958, 745, 835, 844, 719, 424, 419, 21, 933, 168, 538, 537} \[ -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {16 b c e \left (1-c^2 x^2\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{5 c d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {4 b e \left (1-c^2 x^2\right )}{15 c d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) (d+e x)^{3/2}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 d x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {16 b c^2 \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right )^2 \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {1-c^2 x^2} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {1-c^2 x^2} \sqrt {\frac {c (d+e x)}{c d+e}} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c d^2 e x \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 168
Rule 419
Rule 424
Rule 537
Rule 538
Rule 719
Rule 745
Rule 835
Rule 844
Rule 933
Rule 958
Rule 1574
Rule 5227
Rubi steps
\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx &=-\frac {2 \left (a+b \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-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 \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {\left (4 b \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 c d^2 \left (d^2-\frac {e^2}{c^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-c x} \sqrt {1+c 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 \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b \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 c d \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{5 c d^2 \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 ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {d+\frac {e}{c}-\frac {e x^2}{c}}} \, dx,x,\sqrt {1-c 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 \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}+\frac {\left (8 b \sqrt {-\frac {1}{c^2}+x^2}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {-\frac {1}{c^2}+x^2}} \, dx}{15 c \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (2 b \left (-d^2+\frac {e^2}{c^2}\right ) \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 )^2 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {\left (4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {2-x^2} \sqrt {1-\frac {e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\sqrt {1-c x}\right )}{5 c d^2 e \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {\left (4 b \sqrt {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{5 c^2 d^2 \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 {e}{c}}}}\\ &=\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 \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c 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 {d+e x} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 \left (d^2-\frac {e^2}{c^2}\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {d+e x}{d+\frac {e}{c}}}}-\frac {\left (4 b \left (-d^2+\frac {e^2}{c^2}\right ) \sqrt {\frac {d+e x}{d+\frac {e}{c}}} \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1-\frac {2 e x^2}{c \left (d+\frac {e}{c}\right )}}} \, dx,x,\frac {\sqrt {1-c x}}{\sqrt {2}}\right )}{15 c^2 d \left (d^2-\frac {e^2}{c^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 )}{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 \csc ^{-1}(c x)\right )}{5 e (d+e x)^{5/2}}-\frac {16 b c^2 \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 \left (c^2 d^2-e^2\right )^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 d^2 \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {4 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{15 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 {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c 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 [A] time = 13.99, size = 1002, normalized size = 1.86 \[ \frac {b \left (\frac {2 \left (\frac {d}{x}+e\right )^{7/2} (c x)^{7/2} \left (\frac {2 \left (c^2 d^2 e-e^3\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}+\frac {2 \left (3 c^3 d^3+c e^2 d\right ) \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} (c x)^{3/2}}+\frac {2 \left (3 e^3-7 c^2 d^2 e\right ) \cos \left (2 \csc ^{-1}(c x)\right ) \left (d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right ) c^2-\frac {x (c x+1) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e F\left (\sin ^{-1}\left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )\right ) c}{\sqrt {\frac {e (c x+1)}{e-c d}}}+e x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \Pi \left (2;\sin ^{-1}\left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right ) c+(c d+c e x) \left (c^2 x^2-1\right )\right )}{c d \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {\frac {d}{x}+e} \sqrt {c x} \left (c^2 x^2-2\right )}\right )}{15 c d (c d-e)^2 e (c d+e)^2 (d+e x)^{7/2}}-\frac {c^4 \left (\frac {d}{x}+e\right )^4 x^4 \left (-\frac {2 \csc ^{-1}(c x) e^2}{5 c^3 d^3 \left (\frac {d}{x}+e\right )^3}-\frac {2 \left (9 \csc ^{-1}(c x) e^3+2 c d \sqrt {1-\frac {1}{c^2 x^2}} e^2-9 c^2 d^2 \csc ^{-1}(c x) e\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right ) \left (\frac {d}{x}+e\right )^2}-\frac {2 \left (9 c^4 \csc ^{-1}(c x) d^4-16 c^3 e \sqrt {1-\frac {1}{c^2 x^2}} d^3-18 c^2 e^2 \csc ^{-1}(c x) d^2+8 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} d+9 e^4 \csc ^{-1}(c x)\right )}{15 c^3 d^3 \left (c^2 d^2-e^2\right )^2 \left (\frac {d}{x}+e\right )}+\frac {4 \left (3 e^2-7 c^2 d^2\right ) \sqrt {1-\frac {1}{c^2 x^2}}}{15 c^2 d^2 \left (e^2-c^2 d^2\right )^2}+\frac {2 \csc ^{-1}(c x)}{5 c^3 d^3 e}\right )}{(d+e x)^{7/2}}\right )}{c}-\frac {2 a}{5 e (d+e x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 1640, normalized size = 3.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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