Optimal. Leaf size=191 \[ -\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{32 \sqrt {2} c^{3/2} d^{7/2} e}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}+\frac {15 \sqrt {d+e x}}{32 c d^3 e \sqrt {c d^2-c e^2 x^2}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {673, 667, 661, 208} \begin {gather*} -\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{32 \sqrt {2} c^{3/2} d^{7/2} e}+\frac {15 \sqrt {d+e x}}{32 c d^3 e \sqrt {c d^2-c e^2 x^2}}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 208
Rule 661
Rule 667
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}+\frac {5 \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx}{8 d}\\ &=-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {15 \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx}{32 d^2}\\ &=-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {15 \sqrt {d+e x}}{32 c d^3 e \sqrt {c d^2-c e^2 x^2}}+\frac {15 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{64 c d^3}\\ &=-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {15 \sqrt {d+e x}}{32 c d^3 e \sqrt {c d^2-c e^2 x^2}}+\frac {(15 e) \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 )}{32 c d^3}\\ &=-\frac {1}{4 c d e (d+e x)^{3/2} \sqrt {c d^2-c e^2 x^2}}-\frac {5}{16 c d^2 e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {15 \sqrt {d+e x}}{32 c d^3 e \sqrt {c d^2-c e^2 x^2}}-\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{32 \sqrt {2} c^{3/2} d^{7/2} e}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 143, normalized size = 0.75 \begin {gather*} \frac {2 \sqrt {d} \sqrt {d+e x} \left (-3 d^2+20 d e x+15 e^2 x^2\right )-15 \sqrt {2} (d+e x)^2 \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{64 c d^{7/2} e (d+e x)^2 \sqrt {c \left (d^2-e^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 1.93, size = 159, normalized size = 0.83 \begin {gather*} \frac {15 \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 )}{32 \sqrt {2} c^{3/2} d^{7/2} e}+\frac {\left (-8 d^2-10 d (d+e x)+15 (d+e x)^2\right ) \sqrt {2 c d (d+e x)-c (d+e x)^2}}{32 c^2 d^3 e (d-e x) (d+e x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 412, normalized size = 2.16 \begin {gather*} \left [\frac {15 \, \sqrt {2} {\left (e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (15 \, d e^{2} x^{2} + 20 \, d^{2} e x - 3 \, d^{3}\right )} \sqrt {e x + d}}{128 \, {\left (c^{2} d^{4} e^{5} x^{4} + 2 \, c^{2} d^{5} e^{4} x^{3} - 2 \, c^{2} d^{7} e^{2} x - c^{2} d^{8} e\right )}}, -\frac {15 \, \sqrt {2} {\left (e^{4} x^{4} + 2 \, d e^{3} x^{3} - 2 \, d^{3} e x - d^{4}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (15 \, d e^{2} x^{2} + 20 \, d^{2} e x - 3 \, d^{3}\right )} \sqrt {e x + d}}{64 \, {\left (c^{2} d^{4} e^{5} x^{4} + 2 \, c^{2} d^{5} e^{4} x^{3} - 2 \, c^{2} d^{7} e^{2} x - c^{2} d^{8} e\right )}}\right ] \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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 217, normalized size = 1.14 \begin {gather*} \frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (15 \sqrt {-\left (e x -d \right ) c}\, \sqrt {2}\, e^{2} x^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+30 \sqrt {-\left (e x -d \right ) c}\, \sqrt {2}\, d e x \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )-30 \sqrt {c d}\, e^{2} x^{2}+15 \sqrt {-\left (e x -d \right ) c}\, \sqrt {2}\, d^{2} \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )-40 \sqrt {c d}\, d e x +6 \sqrt {c d}\, d^{2}\right )}{64 \left (e x +d \right )^{\frac {5}{2}} \left (e x -d \right ) \sqrt {c d}\, c^{2} d^{3} 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 {1}{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{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 {1}{{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^{3/2}} \,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 {1}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________