Optimal. Leaf size=207 \[ \frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2} \]
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Rubi [A] time = 0.27, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {862, 872, 874, 205} \begin {gather*} \frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \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 f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 862
Rule 872
Rule 874
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^3} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {(c d) \int \frac {\sqrt {d+e x}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 g}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 g (c d f-a e g)}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (c^2 d^2 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g 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 )}{4 g (c d f-a e g)}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {c^2 d^2 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d f-a e g} \sqrt {d+e x}}\right )}{4 g^{3/2} (c d f-a e g)^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 79, normalized size = 0.38 \begin {gather*} \frac {2 c^2 d^2 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{3 (d+e x)^{3/2} (c d f-a e g)^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 180.01, 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 = 1056, normalized size = 5.10 \begin {gather*} \left [\frac {{\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {-c d f g + a e g^{2}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x + 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d f g + a e g^{2}} \sqrt {e x + d}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) - 2 \, {\left (c^{2} d^{2} f^{2} g - 3 \, a c d e f g^{2} + 2 \, a^{2} e^{2} g^{3} - {\left (c^{2} d^{2} f g^{2} - a c d e g^{3}\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}}{8 \, {\left (c^{2} d^{3} f^{4} g^{2} - 2 \, a c d^{2} e f^{3} g^{3} + a^{2} d e^{2} f^{2} g^{4} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{4}\right )} x\right )}}, -\frac {{\left (c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + {\left (2 \, c^{2} d^{2} e f g + c^{2} d^{3} g^{2}\right )} x^{2} + {\left (c^{2} d^{2} e f^{2} + 2 \, c^{2} d^{3} f g\right )} x\right )} \sqrt {c d f g - a e g^{2}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d f g - a e g^{2}} \sqrt {e x + d}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + {\left (c^{2} d^{2} f^{2} g - 3 \, a c d e f g^{2} + 2 \, a^{2} e^{2} g^{3} - {\left (c^{2} d^{2} f g^{2} - a c d e g^{3}\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}}{4 \, {\left (c^{2} d^{3} f^{4} g^{2} - 2 \, a c d^{2} e f^{3} g^{3} + a^{2} d e^{2} f^{2} g^{4} + {\left (c^{2} d^{2} e f^{2} g^{4} - 2 \, a c d e^{2} f g^{5} + a^{2} e^{3} g^{6}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g^{3} + a^{2} d e^{2} g^{6} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{4} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} g^{2} + 2 \, a^{2} d e^{2} f g^{5} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g^{3} - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{4}\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 = 0.03, size = 285, normalized size = 1.38 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (c^{2} d^{2} g^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+2 c^{2} d^{2} f g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+c^{2} d^{2} f^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d g x -2 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a e g +\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c d f \right )}{4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) \left (g x +f \right )^{2} \sqrt {\left (a e g -c d f \right ) g}\, g} \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}}{\sqrt {e x + d} {\left (g x + f\right )}^{3}}\,{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 (f+g\,x\right )}^3\,\sqrt {d+e\,x}} \,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 {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt {d + e x} \left (f + g x\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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