Optimal. Leaf size=280 \[ \frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 \sqrt {g} (c d f-a e g)^{7/2}}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \]
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Rubi [A] time = 0.42, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {872, 874, 205} \begin {gather*} \frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^3 d^3 \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 )}{8 \sqrt {g} (c d f-a e g)^{7/2}}+\frac {5 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 872
Rule 874
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {(5 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 (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}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {\left (5 c^2 d^2\right ) \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}{8 (c d f-a e g)^2}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^3 d^3\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}{16 (c d f-a e g)^3}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^3 d^3 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 )}{8 (c d f-a e g)^3}\\ &=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^3 d^3 \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 )}{8 \sqrt {g} (c d f-a e g)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 77, normalized size = 0.28 \begin {gather*} \frac {2 c^3 d^3 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{\sqrt {d+e x} (c d f-a e g)^4} \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 = 0.47, size = 2027, normalized size = 7.24
result too large to display
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 = 0.02, size = 450, normalized size = 1.61 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{3} d^{3} g^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+45 c^{3} d^{3} f \,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 )+45 c^{3} d^{3} f^{2} g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+15 c^{3} d^{3} f^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x -40 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x -8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+26 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g -33 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right )^{3} \left (g x +f \right )^{3} \sqrt {\left (a e g -c d f \right ) 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 {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}}\,{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 {d+e\,x}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,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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