Optimal. Leaf size=139 \[ -\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} \sqrt {d} e}+\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {663, 661, 208} \begin {gather*} -\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} \sqrt {d} e}+\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 661
Rule 663
Rubi steps
\begin {align*} \int \frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx &=-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac {1}{4} (3 c) \int \frac {\sqrt {c d^2-c e^2 x^2}}{(d+e x)^{5/2}} \, dx\\ &=\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {1}{8} \left (3 c^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx\\ &=\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}+\frac {1}{4} \left (3 c^2 e\right ) \operatorname {Subst}\left (\int \frac {1}{-2 c d e^2+e^2 x^2} \, dx,x,\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {d+e x}}\right )\\ &=\frac {3 c \sqrt {c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac {\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}}-\frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} \sqrt {d} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 109, normalized size = 0.78 \begin {gather*} \frac {c \sqrt {c \left (d^2-e^2 x^2\right )} \left (\frac {2 (d+5 e x)}{(d+e x)^{5/2}}-\frac {3 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {d} \sqrt {d^2-e^2 x^2}}\right )}{8 e} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.36, size = 134, normalized size = 0.96 \begin {gather*} \frac {3 c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sqrt {2 c d (d+e x)-c (d+e x)^2}}{\sqrt {c} (e x-d) \sqrt {d+e x}}\right )}{4 \sqrt {2} \sqrt {d} e}-\frac {c (4 d-5 (d+e x)) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{4 e (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 367, normalized size = 2.64 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt {\frac {c}{d}} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {\frac {c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (5 \, c e x + c d\right )} \sqrt {e x + d}}{8 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}, -\frac {3 \, \sqrt {\frac {1}{2}} {\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt {-\frac {c}{d}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {e x + d} d \sqrt {-\frac {c}{d}}}{c e^{2} x^{2} - c d^{2}}\right ) - \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (5 \, c e x + c d\right )} \sqrt {e x + d}}{4 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] 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]
________________________________________________________________________________________
maple [A] time = 0.07, size = 190, normalized size = 1.37 \begin {gather*} -\frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (3 \sqrt {2}\, c \,e^{2} x^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+6 \sqrt {2}\, c d e x \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+3 \sqrt {2}\, c \,d^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )-10 \sqrt {c d}\, \sqrt {-\left (e x -d \right ) c}\, e x -2 \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, d \right ) c}{8 \left (e x +d \right )^{\frac {5}{2}} \sqrt {-\left (e x -d \right ) c}\, \sqrt {c d}\, e} \end {gather*}
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*} \int \frac {{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________