Optimal. Leaf size=170 \[ -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)} \]
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Rubi [A] time = 0.23, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {878, 874, 205} \[ -\frac {\left (2 a e^2 g-c d (d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} (c d f-a e g)^{3/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{g \sqrt {d+e x} (f+g x) (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 874
Rule 878
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {3}{2} c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\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}{g \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (e^2 \left (2 a e^2 g-c d (e f+d g)\right )\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 )}{g (c d f-a e g)}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{g (c d f-a e g) \sqrt {d+e x} (f+g x)}-\frac {\left (2 a e^2 g-c d (e f+d g)\right ) \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 )}{g^{3/2} (c d f-a e g)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 155, normalized size = 0.91 \[ \frac {\sqrt {d+e x} \left (-\frac {\sqrt {a e+c d x} \left (c d (d g+e f)-2 a e^2 g\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {c d f-a e g}}-\frac {\sqrt {g} (d g-e f) (a e+c d x)}{f+g x}\right )}{g^{3/2} \sqrt {(d+e x) (a e+c d x)} (a e g-c d f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 896, normalized size = 5.27 \[ \left [-\frac {{\left (c d^{2} e f^{2} + {\left (c d^{3} - 2 \, a d e^{2}\right )} f g + {\left (c d e^{2} f g + {\left (c d^{2} e - 2 \, a e^{3}\right )} g^{2}\right )} x^{2} + {\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - a e^{3}\right )} f g + {\left (c d^{3} - 2 \, a d e^{2}\right )} g^{2}\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 d e f^{2} g + a d e g^{3} - {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{2 \, {\left (c^{2} d^{3} f^{3} g^{2} - 2 \, a c d^{2} e f^{2} g^{3} + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{4}\right )} x\right )}}, -\frac {{\left (c d^{2} e f^{2} + {\left (c d^{3} - 2 \, a d e^{2}\right )} f g + {\left (c d e^{2} f g + {\left (c d^{2} e - 2 \, a e^{3}\right )} g^{2}\right )} x^{2} + {\left (c d e^{2} f^{2} + 2 \, {\left (c d^{2} e - a e^{3}\right )} f g + {\left (c d^{3} - 2 \, a d e^{2}\right )} g^{2}\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 d e f^{2} g + a d e g^{3} - {\left (c d^{2} + a e^{2}\right )} f g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c^{2} d^{3} f^{3} g^{2} - 2 \, a c d^{2} e f^{2} g^{3} + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{2} e f^{2} g^{3} - 2 \, a c d e^{2} f g^{4} + a^{2} e^{3} g^{5}\right )} x^{2} + {\left (c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f g^{4}\right )} x}\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 [B] time = 0.03, size = 347, normalized size = 2.04 \[ \frac {\left (-2 a \,e^{2} g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+c \,d^{2} g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+c d e f g x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-2 a \,e^{2} f g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+c \,d^{2} f g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )+c d e \,f^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, d g +\sqrt {\left (a e g -c d f \right ) g}\, \sqrt {c d x +a e}\, e f \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \left (a e g -c d f \right ) \left (g x +f \right ) \sqrt {\left (a e g -c d f \right ) g}\, g} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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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