Optimal. Leaf size=120 \[ -\frac {2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac {2 c d}{\sqrt {d+e x} \left (c d^2-a e^2\right )^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \begin {gather*} -\frac {2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac {2 c d}{\sqrt {d+e x} \left (c d^2-a e^2\right )^2}+\frac {2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 626
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac {1}{(a e+c d x) (d+e x)^{5/2}} \, dx\\ &=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {(c d) \int \frac {1}{(a e+c d x) (d+e x)^{3/2}} \, dx}{c d^2-a e^2}\\ &=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {2 c d}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{\left (c d^2-a e^2\right )^2}\\ &=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {2 c d}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (2 c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{3 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}+\frac {2 c d}{\left (c d^2-a e^2\right )^2 \sqrt {d+e x}}-\frac {2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.48 \begin {gather*} \frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c d (d+e x)}{c d^2-a e^2}\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 128, normalized size = 1.07 \begin {gather*} \frac {2 \left (-a e^2+c d^2+3 c d (d+e x)\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac {2 c^{3/2} d^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x} \sqrt {a e^2-c d^2}}{c d^2-a e^2}\right )}{\left (a e^2-c d^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 461, normalized size = 3.84 \begin {gather*} \left [\frac {3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {\frac {c d}{c d^{2} - a e^{2}}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 2 \, {\left (3 \, c d e x + 4 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{3 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}}, -\frac {2 \, {\left (3 \, {\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac {{\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - {\left (3 \, c d e x + 4 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, {\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} + {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \, {\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 0.06, size = 117, normalized size = 0.98 \begin {gather*} \frac {2 c^{2} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}+\frac {2 c d}{\left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {e x +d}}-\frac {2}{3 \left (a \,e^{2}-c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.69, size = 127, normalized size = 1.06 \begin {gather*} \frac {2\,c^{3/2}\,d^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,\sqrt {d+e\,x}\,\left (a^2\,e^4-2\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}}\right )}{{\left (a\,e^2-c\,d^2\right )}^{5/2}}-\frac {\frac {2}{3\,\left (a\,e^2-c\,d^2\right )}-\frac {2\,c\,d\,\left (d+e\,x\right )}{{\left (a\,e^2-c\,d^2\right )}^2}}{{\left (d+e\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.82, size = 107, normalized size = 0.89 \begin {gather*} \frac {2 c d}{\sqrt {d + e x} \left (a e^{2} - c d^{2}\right )^{2}} + \frac {2 c d \operatorname {atan}{\left (\frac {\sqrt {d + e x}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}}} \right )}}{\sqrt {\frac {a e^{2} - c d^{2}}{c d}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac {2}{3 \left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} - c d^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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