Optimal. Leaf size=374 \[ -\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{35 e^2 \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}-\frac {b x \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {c^2 x^2}}-\frac {b x \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}} \]
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Rubi [A] time = 0.54, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {266, 43, 5239, 12, 573, 154, 157, 63, 217, 206, 93, 204} \[ -\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {b x \sqrt {c^2 x^2-1} \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {b x \left (-35 c^4 d^2 e+35 c^6 d^3-63 c^2 d e^2-25 e^3\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \left (13 c^2 d+25 e\right ) \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 63
Rule 93
Rule 154
Rule 157
Rule 204
Rule 206
Rule 217
Rule 266
Rule 573
Rule 5239
Rubi steps
\begin {align*} \int x^3 \left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{35 e^2 x \sqrt {-1+c^2 x^2}} \, dx}{\sqrt {c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{5/2} \left (-2 d+5 e x^2\right )}{x \sqrt {-1+c^2 x^2}} \, dx}{35 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b c x) \operatorname {Subst}\left (\int \frac {(d+e x)^{5/2} (-2 d+5 e x)}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{70 e^2 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b x) \operatorname {Subst}\left (\int \frac {(d+e x)^{3/2} \left (-6 c^2 d^2+\frac {1}{2} e \left (13 c^2 d+25 e\right ) x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{210 c e^2 \sqrt {c^2 x^2}}\\ &=\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b x) \operatorname {Subst}\left (\int \frac {\sqrt {d+e x} \left (-12 c^4 d^3-\frac {3}{4} e \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x\right )}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{420 c^3 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {(b x) \operatorname {Subst}\left (\int \frac {-12 c^6 d^4-\frac {3}{8} e \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{420 c^5 e^2 \sqrt {c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {\left (b c d^4 x\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{1120 c^5 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}-\frac {\left (2 b c d^4 x\right ) \operatorname {Subst}\left (\int \frac {1}{-d-x^2} \, dx,x,\frac {\sqrt {d+e x^2}}{\sqrt {-1+c^2 x^2}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x\right ) \operatorname {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 )}{560 c^7 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {\left (b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x\right ) \operatorname {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 )}{560 c^7 e \sqrt {c^2 x^2}}\\ &=-\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {2 b c d^{7/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.62, size = 339, normalized size = 0.91 \[ \frac {\sqrt {d+e x^2} \left (-48 a c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2-48 b c^5 \csc ^{-1}(c x) \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2+b e x \sqrt {1-\frac {1}{c^2 x^2}} \left (c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )+2 c^2 e \left (82 d+25 e x^2\right )+75 e^2\right )\right )}{1680 c^5 e^2}+\frac {b \left (32 c^4 d^4 \left (c^2 x^2-1\right ) \sqrt {\frac {d}{e x^2}+1} F_1\left (1;\frac {1}{2},\frac {1}{2};2;\frac {1}{c^2 x^2},-\frac {d}{e x^2}\right )+e x^4 \sqrt {1-\frac {1}{c^2 x^2}} \sqrt {1-c^2 x^2} \left (-35 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2+25 e^3\right ) \sqrt {\frac {e x^2}{d}+1} F_1\left (1;\frac {1}{2},\frac {1}{2};2;c^2 x^2,-\frac {e x^2}{d}\right )\right )}{1120 c^5 e^2 x \left (c^2 x^2-1\right ) \sqrt {d+e x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 9.25, size = 1697, normalized size = 4.54 \[ \text {result too large to display} \]
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 \[ \int {\left (e x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 6.43, size = 0, normalized size = 0.00 \[ \int x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx \]
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 \[ \frac {1}{35} \, {\left (\frac {5 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{e} - \frac {2 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d}{e^{2}}\right )} a + \frac {{\left (e^{2} \int \frac {{\left (5 \, c^{2} e^{3} x^{7} + 8 \, c^{2} d e^{2} x^{5} + c^{2} d^{2} e x^{3} - 2 \, c^{2} d^{3} x\right )} e^{\left (\frac {1}{2} \, \log \left (e x^{2} + d\right ) + \frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (c x - 1\right )\right )}}{c^{2} e^{2} x^{2} + {\left (c^{2} e^{2} x^{2} - e^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} - e^{2}}\,{d x} + {\left (5 \, e^{3} x^{6} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 8 \, d e^{2} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + d^{2} e x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 2 \, d^{3} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} \sqrt {e x^{2} + d}\right )} b}{35 \, e^{2}} \]
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 \[ \int x^3\,{\left (e\,x^2+d\right )}^{3/2}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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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