Optimal. Leaf size=150 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} c^{3/2} d^{5/2} e}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {673, 667, 661, 208} \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c d^2-c e^2 x^2}}{\sqrt {2} \sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{4 \sqrt {2} c^{3/2} d^{5/2} e}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 667
Rule 673
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}} \, dx &=-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \int \frac {\sqrt {d+e x}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx}{4 d}\\ &=-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}+\frac {3 \int \frac {1}{\sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}} \, dx}{8 c d^2}\\ &=-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}+\frac {(3 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 )}{4 c d^2}\\ &=-\frac {1}{2 c d e \sqrt {d+e x} \sqrt {c d^2-c e^2 x^2}}+\frac {3 \sqrt {d+e x}}{4 c d^2 e \sqrt {c d^2-c e^2 x^2}}-\frac {3 \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} c^{3/2} d^{5/2} e}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 128, normalized size = 0.85 \[ \frac {2 \sqrt {d} \sqrt {d+e x} (d+3 e x)-3 \sqrt {2} (d+e x) \sqrt {d^2-e^2 x^2} \tanh ^{-1}\left (\frac {\sqrt {d^2-e^2 x^2}}{\sqrt {2} \sqrt {d} \sqrt {d+e x}}\right )}{8 c d^{5/2} e (d+e x) \sqrt {c \left (d^2-e^2 x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 382, normalized size = 2.55 \[ \left [\frac {3 \, \sqrt {2} {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt {c d} \log \left (-\frac {c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {c d} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 4 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (3 \, d e x + d^{2}\right )} \sqrt {e x + d}}{16 \, {\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}, -\frac {3 \, \sqrt {2} {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \sqrt {-c d} \arctan \left (\frac {\sqrt {2} \sqrt {-c e^{2} x^{2} + c d^{2}} \sqrt {-c d} \sqrt {e x + d}}{c e^{2} x^{2} - c d^{2}}\right ) + 2 \, \sqrt {-c e^{2} x^{2} + c d^{2}} {\left (3 \, d e x + d^{2}\right )} \sqrt {e x + d}}{8 \, {\left (c^{2} d^{3} e^{4} x^{3} + c^{2} d^{4} e^{3} x^{2} - c^{2} d^{5} e^{2} x - c^{2} d^{6} e\right )}}\right ] \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 152, normalized size = 1.01 \[ \frac {\sqrt {-\left (e^{2} x^{2}-d^{2}\right ) c}\, \left (3 \sqrt {-\left (e x -d \right ) c}\, \sqrt {2}\, e x \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )+3 \sqrt {-\left (e x -d \right ) c}\, \sqrt {2}\, d \arctanh \left (\frac {\sqrt {-\left (e x -d \right ) c}\, \sqrt {2}}{2 \sqrt {c d}}\right )-6 \sqrt {c d}\, e x -2 \sqrt {c d}\, d \right )}{8 \left (e x +d \right )^{\frac {3}{2}} \left (e x -d \right ) \sqrt {c d}\, c^{2} d^{2} e} \]
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 \[ \int \frac {1}{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac {3}{2}} \sqrt {e x + d}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,d^2-c\,e^2\,x^2\right )}^{3/2}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {1}{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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