Optimal. Leaf size=374 \[ -\frac {b \left (3 c^4 d^2-38 c^2 d e-25 e^2\right ) x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{560 c^5 e \sqrt {c^2 x^2}}+\frac {b \left (13 c^2 d+25 e\right ) x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{3/2}}{840 c^3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {-1+c^2 x^2} \left (d+e x^2\right )^{5/2}}{42 c e \sqrt {c^2 x^2}}-\frac {d \left (d+e x^2\right )^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^2}+\frac {\left (d+e x^2\right )^{7/2} \left (a+b \csc ^{-1}(c x)\right )}{7 e^2}+\frac {2 b c d^{7/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{35 e^2 \sqrt {c^2 x^2}}-\frac {b \left (35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{560 c^6 e^{3/2} \sqrt {c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.36, 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 = {272, 45,
5347, 12, 587, 159, 163, 65, 223, 212, 95, 210} \begin {gather*} -\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 \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 65
Rule 95
Rule 159
Rule 163
Rule 210
Rule 212
Rule 223
Rule 272
Rule 587
Rule 5347
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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \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 )}{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 ) \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 )}{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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.33, size = 295, normalized size = 0.79 \begin {gather*} \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+b e \sqrt {1-\frac {1}{c^2 x^2}} x \left (75 e^2+2 c^2 e \left (82 d+25 e x^2\right )+c^4 \left (57 d^2+106 d e x^2+40 e^2 x^4\right )\right )-48 b c^5 \left (2 d-5 e x^2\right ) \left (d+e x^2\right )^2 \csc ^{-1}(c x)\right )}{1680 c^5 e^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (-32 c^7 d^{7/2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \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 {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{560 c^6 e^2 \sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.37, size = 865, normalized size = 2.31 \begin {gather*} \left [\frac {{\left (96 \, b c^{7} \sqrt {-d} d^{3} \log \left (\frac {c^{4} d^{2} x^{4} - 8 \, c^{2} d^{2} x^{2} + x^{4} e^{2} - 4 \, {\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {-d} + 8 \, d^{2} - 2 \, {\left (3 \, c^{2} d x^{4} - 4 \, d x^{2}\right )} e}{x^{4}}\right ) - 3 \, {\left (35 \, b c^{6} d^{3} - 35 \, b c^{4} d^{2} e - 63 \, b c^{2} d e^{2} - 25 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (c^{4} d^{2} + 4 \, {\left (c^{3} d + {\left (2 \, c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + {\left (8 \, c^{4} x^{4} - 8 \, c^{2} x^{2} + 1\right )} e^{2} + 2 \, {\left (4 \, c^{4} d x^{2} - 3 \, c^{2} d\right )} e\right ) + 4 \, {\left (240 \, a c^{7} x^{6} e^{3} + 384 \, a c^{7} d x^{4} e^{2} + 48 \, a c^{7} d^{2} x^{2} e - 96 \, a c^{7} d^{3} + 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (57 \, b c^{5} d^{2} e + 5 \, {\left (8 \, b c^{5} x^{4} + 10 \, b c^{3} x^{2} + 15 \, b c\right )} e^{3} + 2 \, {\left (53 \, b c^{5} d x^{2} + 82 \, b c^{3} d\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{6720 \, c^{7}}, \frac {{\left (192 \, b c^{7} d^{\frac {7}{2}} \arctan \left (-\frac {{\left (c^{2} d x^{2} - x^{2} e - 2 \, d\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} \sqrt {d}}{2 \, {\left (c^{2} d^{2} x^{2} - d^{2} + {\left (c^{2} d x^{4} - d x^{2}\right )} e\right )}}\right ) - 3 \, {\left (35 \, b c^{6} d^{3} - 35 \, b c^{4} d^{2} e - 63 \, b c^{2} d e^{2} - 25 \, b e^{3}\right )} e^{\frac {1}{2}} \log \left (c^{4} d^{2} + 4 \, {\left (c^{3} d + {\left (2 \, c^{3} x^{2} - c\right )} e\right )} \sqrt {c^{2} x^{2} - 1} \sqrt {x^{2} e + d} e^{\frac {1}{2}} + {\left (8 \, c^{4} x^{4} - 8 \, c^{2} x^{2} + 1\right )} e^{2} + 2 \, {\left (4 \, c^{4} d x^{2} - 3 \, c^{2} d\right )} e\right ) + 4 \, {\left (240 \, a c^{7} x^{6} e^{3} + 384 \, a c^{7} d x^{4} e^{2} + 48 \, a c^{7} d^{2} x^{2} e - 96 \, a c^{7} d^{3} + 48 \, {\left (5 \, b c^{7} x^{6} e^{3} + 8 \, b c^{7} d x^{4} e^{2} + b c^{7} d^{2} x^{2} e - 2 \, b c^{7} d^{3}\right )} \operatorname {arccsc}\left (c x\right ) + {\left (57 \, b c^{5} d^{2} e + 5 \, {\left (8 \, b c^{5} x^{4} + 10 \, b c^{3} x^{2} + 15 \, b c\right )} e^{3} + 2 \, {\left (53 \, b c^{5} d x^{2} + 82 \, b c^{3} d\right )} e^{2}\right )} \sqrt {c^{2} x^{2} - 1}\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-2\right )}}{6720 \, c^{7}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________