Optimal. Leaf size=264 \[ \frac {c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {662, 672, 660, 205} \begin {gather*} \frac {c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac {c^3 d^3 \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 )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 e (d+e x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 660
Rule 662
Rule 672
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{9/2}} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {(c d) \int \frac {1}{(d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 e}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {\left (c^2 d^2\right ) \int \frac {1}{(d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 e \left (c d^2-a e^2\right )}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\right ) \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}{16 e \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {\left (c^3 d^3\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 )}{8 \left (c d^2-a e^2\right )^2}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 e (d+e x)^{7/2}}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}+\frac {c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}+\frac {c^3 d^3 \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 )}{8 e^{3/2} \left (c d^2-a e^2\right )^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 83, normalized size = 0.31 \begin {gather*} \frac {2 c^3 d^3 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (\frac {3}{2},4;\frac {5}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.12, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.45, size = 1115, normalized size = 4.22 \begin {gather*} \left [-\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {-c d^{2} e + a e^{3}} \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} \sqrt {-c d^{2} e + a e^{3}} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (3 \, c^{3} d^{6} e - 17 \, a c^{2} d^{4} e^{3} + 22 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} - 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{48 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x\right )}}, -\frac {3 \, {\left (c^{3} d^{3} e^{4} x^{4} + 4 \, c^{3} d^{4} e^{3} x^{3} + 6 \, c^{3} d^{5} e^{2} x^{2} + 4 \, c^{3} d^{6} e x + c^{3} d^{7}\right )} \sqrt {c d^{2} e - a e^{3}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d^{2} e - a e^{3}} \sqrt {e x + d}}{c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x}\right ) + {\left (3 \, c^{3} d^{6} e - 17 \, a c^{2} d^{4} e^{3} + 22 \, a^{2} c d^{2} e^{5} - 8 \, a^{3} e^{7} - 3 \, {\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} x^{2} - 2 \, {\left (4 \, c^{3} d^{5} e^{2} - 5 \, a c^{2} d^{3} e^{4} + a^{2} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{24 \, {\left (c^{3} d^{10} e^{2} - 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{6} e^{6} - a^{3} d^{4} e^{8} + {\left (c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{5} - 3 \, a c^{2} d^{5} e^{7} + 3 \, a^{2} c d^{3} e^{9} - a^{3} d e^{11}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{4} - 3 \, a c^{2} d^{6} e^{6} + 3 \, a^{2} c d^{4} e^{8} - a^{3} d^{2} e^{10}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e^{3} - 3 \, a c^{2} d^{7} e^{5} + 3 \, a^{2} c d^{5} e^{7} - a^{3} d^{3} e^{9}\right )} x\right )}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 457, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (3 c^{3} d^{3} e^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+9 c^{3} d^{4} e^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+9 c^{3} d^{5} e x \arctanh \left (\frac {\sqrt {c d x +a e}\, e}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}}\right )+3 c^{3} d^{6} \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}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{2} e^{2} x^{2}+2 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, a c d \,e^{3} x -8 \sqrt {c d x +a e}\, \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, c^{2} d^{3} e x +8 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a^{2} e^{4}-14 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, a c \,d^{2} e^{2}+3 \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \sqrt {c d x +a e}\, c^{2} d^{4}\right )}{24 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a \,e^{2}-c \,d^{2}\right ) e}\, \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {c d x +a e}\, 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 {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{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 = -0.00 \begin {gather*} \int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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