Optimal. Leaf size=253 \[ \frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{7/2} \sqrt {c d f-a e g}}-\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3} \]
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Rubi [A] time = 0.34, antiderivative size = 253, 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 = {862, 874, 205} \[ -\frac {5 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 g^3 \sqrt {d+e x} (f+g x)}+\frac {5 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{7/2} \sqrt {c d f-a e g}}-\frac {5 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 862
Rule 874
Rubi steps
\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2} (f+g x)^4} \, dx &=-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {(5 c d) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2} (f+g x)^3} \, dx}{6 g}\\ &=-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\frac {\left (5 c^2 d^2\right ) \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)^2} \, dx}{8 g^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 g^3 \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\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 g^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}}{8 g^3 \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\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 g^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}}{8 g^3 \sqrt {d+e x} (f+g x)}-\frac {5 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{12 g^2 (d+e x)^{3/2} (f+g x)^2}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{3 g (d+e x)^{5/2} (f+g x)^3}+\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 g^{7/2} \sqrt {c d f-a e g}}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 171, normalized size = 0.68 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {15 c^3 d^3 \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {a e+c d x} \sqrt {c d f-a e g}}-\frac {\sqrt {g} \left (8 a^2 e^2 g^2+2 a c d e g (5 f+13 g x)+c^2 d^2 \left (15 f^2+40 f g x+33 g^2 x^2\right )\right )}{(f+g x)^3}\right )}{24 g^{7/2} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 1140, normalized size = 4.51 \[ \left [-\frac {15 \, {\left (c^{3} d^{3} e g^{3} x^{4} + c^{3} d^{4} f^{3} + {\left (3 \, c^{3} d^{3} e f g^{2} + c^{3} d^{4} g^{3}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} e f^{2} g + c^{3} d^{4} f g^{2}\right )} x^{2} + {\left (c^{3} d^{3} e f^{3} + 3 \, c^{3} d^{4} f^{2} 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 (15 \, c^{3} d^{3} f^{3} g - 5 \, a c^{2} d^{2} e f^{2} g^{2} - 2 \, a^{2} c d e^{2} f g^{3} - 8 \, a^{3} e^{3} g^{4} + 33 \, {\left (c^{3} d^{3} f g^{3} - a c^{2} d^{2} e g^{4}\right )} x^{2} + 2 \, {\left (20 \, c^{3} d^{3} f^{2} g^{2} - 7 \, a c^{2} d^{2} e f g^{3} - 13 \, a^{2} c d e^{2} g^{4}\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 d^{2} f^{4} g^{4} - a d e f^{3} g^{5} + {\left (c d e f g^{7} - a e^{2} g^{8}\right )} x^{4} + {\left (3 \, c d e f^{2} g^{6} - a d e g^{8} + {\left (c d^{2} - 3 \, a e^{2}\right )} f g^{7}\right )} x^{3} + 3 \, {\left (c d e f^{3} g^{5} - a d e f g^{7} + {\left (c d^{2} - a e^{2}\right )} f^{2} g^{6}\right )} x^{2} + {\left (c d e f^{4} g^{4} - 3 \, a d e f^{2} g^{6} + {\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g^{5}\right )} x\right )}}, -\frac {15 \, {\left (c^{3} d^{3} e g^{3} x^{4} + c^{3} d^{4} f^{3} + {\left (3 \, c^{3} d^{3} e f g^{2} + c^{3} d^{4} g^{3}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} e f^{2} g + c^{3} d^{4} f g^{2}\right )} x^{2} + {\left (c^{3} d^{3} e f^{3} + 3 \, c^{3} d^{4} f^{2} 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 (15 \, c^{3} d^{3} f^{3} g - 5 \, a c^{2} d^{2} e f^{2} g^{2} - 2 \, a^{2} c d e^{2} f g^{3} - 8 \, a^{3} e^{3} g^{4} + 33 \, {\left (c^{3} d^{3} f g^{3} - a c^{2} d^{2} e g^{4}\right )} x^{2} + 2 \, {\left (20 \, c^{3} d^{3} f^{2} g^{2} - 7 \, a c^{2} d^{2} e f g^{3} - 13 \, a^{2} c d e^{2} g^{4}\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 d^{2} f^{4} g^{4} - a d e f^{3} g^{5} + {\left (c d e f g^{7} - a e^{2} g^{8}\right )} x^{4} + {\left (3 \, c d e f^{2} g^{6} - a d e g^{8} + {\left (c d^{2} - 3 \, a e^{2}\right )} f g^{7}\right )} x^{3} + 3 \, {\left (c d e f^{3} g^{5} - a d e f g^{7} + {\left (c d^{2} - a e^{2}\right )} f^{2} g^{6}\right )} x^{2} + {\left (c d e f^{4} g^{4} - 3 \, a d e f^{2} g^{6} + {\left (3 \, c d^{2} - a e^{2}\right )} f^{3} g^{5}\right )} x\right )}}\right ] \]
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 [A] time = 0.03, size = 441, normalized size = 1.74 \[ -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (15 c^{3} d^{3} g^{3} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+45 c^{3} d^{3} f \,g^{2} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+45 c^{3} d^{3} f^{2} g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+15 c^{3} d^{3} f^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+33 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} g^{2} x^{2}+26 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e \,g^{2} x +40 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f g x +8 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a^{2} e^{2} g^{2}+10 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, a c d e f g +15 \sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, c^{2} d^{2} f^{2}\right )}{24 \sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (g x +f \right )^{3} \sqrt {\left (a e g -c d f \right ) g}\, g^{3}} \]
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 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}}{{\left (f+g\,x\right )}^4\,{\left (d+e\,x\right )}^{5/2}} \,d x \]
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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