Optimal. Leaf size=195 \[ \frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b c d^{3/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e \sqrt {c^2 x^2}}+\frac {b \left (3 c^2 d+e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 \sqrt {e} \sqrt {c^2 x^2}} \]
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Rubi [A]
time = 0.13, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5345, 457, 104,
163, 65, 223, 212, 95, 210} \begin {gather*} \frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b c d^{3/2} x \text {ArcTan}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{3 e \sqrt {c^2 x^2}}+\frac {b x \sqrt {c^2 x^2-1} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {b x \left (3 c^2 d+e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {c^2 x^2-1}}{c \sqrt {d+e x^2}}\right )}{6 c^2 \sqrt {e} \sqrt {c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 104
Rule 163
Rule 210
Rule 212
Rule 223
Rule 457
Rule 5345
Rubi steps
\begin {align*} \int x \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {(b c x) \int \frac {\left (d+e x^2\right )^{3/2}}{x \sqrt {-1+c^2 x^2}} \, dx}{3 e \sqrt {c^2 x^2}}\\ &=\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {(b c x) \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 e \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {(b x) \text {Subst}\left (\int \frac {c^2 d^2+\frac {1}{2} e \left (3 c^2 d+e\right ) x}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 c e \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (b c d^2 x\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{6 e \sqrt {c^2 x^2}}+\frac {\left (b \left (3 c^2 d+e\right ) x\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+c^2 x} \sqrt {d+e x}} \, dx,x,x^2\right )}{12 c \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}+\frac {\left (b c d^2 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 )}{3 e \sqrt {c^2 x^2}}+\frac {\left (b \left (3 c^2 d+e\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 )}{6 c^3 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e \sqrt {c^2 x^2}}+\frac {\left (b \left (3 c^2 d+e\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 )}{6 c^3 \sqrt {c^2 x^2}}\\ &=\frac {b x \sqrt {-1+c^2 x^2} \sqrt {d+e x^2}}{6 c \sqrt {c^2 x^2}}+\frac {\left (d+e x^2\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e}-\frac {b c d^{3/2} x \tan ^{-1}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{3 e \sqrt {c^2 x^2}}+\frac {b \left (3 c^2 d+e\right ) x \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )}{6 c^2 \sqrt {e} \sqrt {c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 196, normalized size = 1.01 \begin {gather*} \frac {\sqrt {d+e x^2} \left (b e \sqrt {1-\frac {1}{c^2 x^2}} x+2 a c \left (d+e x^2\right )+2 b c \left (d+e x^2\right ) \csc ^{-1}(c x)\right )}{6 c e}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (2 c^3 d^{3/2} \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {-1+c^2 x^2}}{\sqrt {d+e x^2}}\right )+\sqrt {e} \left (3 c^2 d+e\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {-1+c^2 x^2}}{c \sqrt {d+e x^2}}\right )\right )}{6 c^2 e \sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 588, normalized size = 3.02 \begin {gather*} \left [\frac {{\left (2 \, b c^{3} \sqrt {-d} d \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 ) + {\left (3 \, b c^{2} d + b e\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 (2 \, a c^{3} x^{2} e + 2 \, a c^{3} d + \sqrt {c^{2} x^{2} - 1} b c e + 2 \, {\left (b c^{3} x^{2} e + b c^{3} d\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}}{24 \, c^{3}}, -\frac {{\left (4 \, b c^{3} d^{\frac {3}{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 ) - {\left (3 \, b c^{2} d + b e\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 (2 \, a c^{3} x^{2} e + 2 \, a c^{3} d + \sqrt {c^{2} x^{2} - 1} b c e + 2 \, {\left (b c^{3} x^{2} e + b c^{3} d\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {x^{2} e + d}\right )} e^{\left (-1\right )}}{24 \, c^{3}}\right ] \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 x \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{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 = -0.01 \begin {gather*} \int x\,\sqrt {e\,x^2+d}\,\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.
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