Optimal. Leaf size=351 \[ -\frac {c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \]
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Rubi [A] time = 0.56, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {878, 872, 874, 205} \[ -\frac {c^2 d^2 \left (6 a e^2 g-c d (5 d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{8 g^{3/2} (c d f-a e g)^{7/2}}-\frac {c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{8 g \sqrt {d+e x} (f+g x) (c d f-a e g)^3}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (6 a e^2 g-c d (5 d g+e f)\right )}{12 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)^2}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 g \sqrt {d+e x} (f+g x)^3 (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 205
Rule 872
Rule 874
Rule 878
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^4 \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}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {11}{2} c d^2 e g-3 e \left (c d^2+a e^2\right ) g\right )\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}{3 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}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {\left (c d \left (6 a e^2 g-c d (e f+5 d g)\right )\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 g (c d f-a e g)^2}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (c^2 d^2 \left (6 a e^2 g-c d (e f+5 d 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}{16 g (c d f-a e g)^3}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {\left (c^2 d^2 e^2 \left (6 a e^2 g-c d (e f+5 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 )}{8 g (c d f-a e g)^3}\\ &=-\frac {(e f-d g) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 g (c d f-a e g) \sqrt {d+e x} (f+g x)^3}-\frac {\left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^2}-\frac {c d \left (6 a e^2 g-c d (e f+5 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{8 g (c d f-a e g)^3 \sqrt {d+e x} (f+g x)}-\frac {c^2 d^2 \left (6 a e^2 g-c d (e f+5 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 )}{8 g^{3/2} (c d f-a e g)^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 132, normalized size = 0.38 \[ \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {e f-d g}{(f+g x)^3}-\frac {c^2 d^2 \left (c d (5 d g+e f)-6 a e^2 g\right ) \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {g (a e+c d x)}{a e g-c d f}\right )}{(c d f-a e g)^3}\right )}{3 g \sqrt {d+e x} (a e g-c d f)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 2736, normalized size = 7.79 \[ \text {result too large to display} \]
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 = 0.05, size = 1142, normalized size = 3.25 \[ -\frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (18 a \,c^{2} d^{2} e^{2} g^{4} x^{3} \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^{4} g^{4} x^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 c^{3} d^{3} e 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 )+54 a \,c^{2} d^{2} e^{2} f \,g^{3} 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^{4} f \,g^{3} x^{2} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-9 c^{3} d^{3} e \,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 )+54 a \,c^{2} d^{2} e^{2} f^{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 )-45 c^{3} d^{4} f^{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 )-9 c^{3} d^{3} e \,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 )+18 a \,c^{2} d^{2} e^{2} f^{3} g \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^{4} f^{3} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 c^{3} d^{3} e \,f^{4} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-18 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} g^{3} x^{2}+15 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} g^{3} x^{2}+3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e f \,g^{2} x^{2}+12 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{3} g^{3} x -10 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c \,d^{2} e \,g^{3} x -50 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} f \,g^{2} x +40 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} f \,g^{2} x +8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e \,f^{2} g x +8 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} d \,e^{2} g^{3}+4 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a^{2} e^{3} f \,g^{2}-26 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c \,d^{2} e f \,g^{2}-16 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a c d \,e^{2} f^{2} g +33 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{3} f^{2} g -3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c^{2} d^{2} e \,f^{3}\right )}{24 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{3} \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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{4}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^4\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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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