Optimal. Leaf size=347 \[ \frac {5 c^4 d^4 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4} \]
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Rubi [A] time = 0.45, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {862, 872, 874, 205} \[ \frac {5 c^4 d^4 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{64 g^{3/2} (c d f-a e g)^{7/2}}+\frac {5 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}+\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{96 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}+\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4} \]
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)^5} \, dx &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {(c d) \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}{8 g}\\ &=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {\left (5 c^2 d^2\right ) \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}{48 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}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {\left (5 c^3 d^3\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}{64 g (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}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^4 d^4\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}{128 g (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}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {\left (5 c^4 d^4 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 )}{64 g (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}}{4 g \sqrt {d+e x} (f+g x)^4}+\frac {c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {5 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{96 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}+\frac {5 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{64 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^4 d^4 \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 )}{64 g^{3/2} (c d f-a e g)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 79, normalized size = 0.23 \[ \frac {2 c^4 d^4 ((d+e x) (a e+c d x))^{3/2} \, _2F_1\left (\frac {3}{2},5;\frac {5}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{3 (d+e x)^{3/2} (c d f-a e g)^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 2610, normalized size = 7.52 \[ \text {result too large to display} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 696, normalized size = 2.01 \[ \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{4} d^{4} g^{4} x^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+60 c^{4} d^{4} f \,g^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+90 c^{4} d^{4} f^{2} g^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+60 c^{4} d^{4} f^{3} g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+15 c^{4} d^{4} f^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} g^{3} x^{3}+10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e \,g^{3} x^{2}-55 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f \,g^{2} x^{2}-8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} c d \,e^{2} g^{3} x +36 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,c^{2} d^{2} e f \,g^{2} x -73 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{3} d^{3} f^{2} g x -48 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{3} e^{3} g^{3}+136 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} c d \,e^{2} f \,g^{2}-118 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a \,c^{2} d^{2} e \,f^{2} g +15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{3} d^{3} f^{3}\right )}{192 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{4} \left (a e g -c d f \right ) \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d x +a e}\, g} \]
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 \[ \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{5}}\,{d x} \]
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 \[ \int \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (f+g\,x\right )}^5\,\sqrt {d+e\,x}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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