Optimal. Leaf size=196 \[ -\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {3 c d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]
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Rubi [A] time = 0.13, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {672, 666, 660, 205} \begin {gather*} -\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{\sqrt {d+e x} \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {3 c d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 660
Rule 666
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(3 c d) \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{2 \left (c d^2-a e^2\right )}\\ &=\frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(3 c d e) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (3 c d e^2\right ) \operatorname {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^2}\\ &=\frac {1}{\left (c d^2-a e^2\right ) \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {d+e x}}{\left (c d^2-a e^2\right )^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {3 c d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{\left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 77, normalized size = 0.39 \begin {gather*} -\frac {2 c d \sqrt {d+e x} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{\left (c d^2-a e^2\right )^2 \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.47, size = 179, normalized size = 0.91 \begin {gather*} \frac {(d+e x)^{3/2} (a e+c d x)^{3/2} \left (-\frac {c d \left (3 e (a e+c d x)-2 a e^2+2 c d^2\right )}{\left (c d^2-a e^2\right )^2 \sqrt {a e+c d x} \left (e (a e+c d x)-a e^2+c d^2\right )}-\frac {3 c d \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}\right )}{((d+e x) (a e+c d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 764, normalized size = 3.90 \begin {gather*} \left [\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a c d^{3} e + {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2}\right )} x\right )} \sqrt {-\frac {e}{c d^{2} - a e^{2}}} \log \left (-\frac {c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {-\frac {e}{c d^{2} - a e^{2}}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{2 \, {\left (a c^{2} d^{6} e - 2 \, a^{2} c d^{4} e^{3} + a^{3} d^{2} e^{5} + {\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{3} + {\left (2 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + a^{3} e^{7}\right )} x^{2} + {\left (c^{3} d^{7} - 3 \, a^{2} c d^{3} e^{4} + 2 \, a^{3} d e^{6}\right )} x\right )}}, -\frac {3 \, {\left (c^{2} d^{2} e^{2} x^{3} + a c d^{3} e + {\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x^{2} + {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2}\right )} x\right )} \sqrt {\frac {e}{c d^{2} - a e^{2}}} \arctan \left (-\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d} \sqrt {\frac {e}{c d^{2} - a e^{2}}}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt {e x + d}}{a c^{2} d^{6} e - 2 \, a^{2} c d^{4} e^{3} + a^{3} d^{2} e^{5} + {\left (c^{3} d^{5} e^{2} - 2 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x^{3} + {\left (2 \, c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + a^{3} e^{7}\right )} x^{2} + {\left (c^{3} d^{7} - 3 \, a^{2} c d^{3} e^{4} + 2 \, a^{3} d e^{6}\right )} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 235, normalized size = 1.20 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (3 \sqrt {c d x +a e}\, c d \,e^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+3 \sqrt {c d x +a e}\, c \,d^{2} e \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )-3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c d e x -\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a \,e^{2}-2 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c \,d^{2}\right )}{\left (e x +d \right )^{\frac {3}{2}} \left (c d x +a e \right ) \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}} \end {gather*}
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*} \int \frac {1}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} \sqrt {e x + d}}\,{d x} \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 \frac {1}{\sqrt {d+e\,x}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \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 \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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