Optimal. Leaf size=261 \[ -\frac {c d \left (4 a e^2 g-c d (3 d g+e f)\right ) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2 (c d f-a e g)}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)^2} \]
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Rubi [A] time = 0.36, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {878, 872, 874, 205} \begin {gather*} -\frac {c d \left (4 a e^2 g-c d (3 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 )}{4 g^{3/2} (c d f-a e g)^{5/2}}-\frac {(e f-d g) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 g \sqrt {d+e x} (f+g x)^2 (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} \left (4 a e^2 g-c d (3 d g+e f)\right )}{4 g \sqrt {d+e x} (f+g x) (c d f-a e g)^2} \end {gather*}
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)^3 \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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}+\frac {\left (e \left (\frac {1}{2} c d e^2 f+\frac {7}{2} c d^2 e g-2 e \left (c d^2+a e^2\right ) 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}{2 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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d \left (4 a e^2 g-c d (e f+3 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}{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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {\left (c d e^2 \left (4 a e^2 g-c d (e f+3 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 )}{4 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}}{2 g (c d f-a e g) \sqrt {d+e x} (f+g x)^2}-\frac {\left (4 a e^2 g-c d (e f+3 d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 g (c d f-a e g)^2 \sqrt {d+e x} (f+g x)}-\frac {c d \left (4 a e^2 g-c d (e f+3 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 )}{4 g^{3/2} (c d f-a e g)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 189, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(d+e x) (a e+c d x)} \left (\frac {c d \left (2 a e^2 g-\frac {1}{2} c d (3 d g+e f)\right ) \left (\frac {c d f-a e g}{c d f+c d g x}+\frac {\sqrt {c d f-a e g} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )}{\sqrt {g} \sqrt {a e+c d x}}\right )}{(c d f-a e g)^2}+\frac {e f-d g}{(f+g x)^2}\right )}{2 g \sqrt {d+e x} (a e g-c d f)} \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 = 1704, normalized size = 6.53
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 [B] time = 0.04, size = 673, normalized size = 2.58 \begin {gather*} \frac {\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (4 a c d \,e^{2} 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 )-3 c^{2} d^{3} 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 )-c^{2} d^{2} e 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 )+8 a c d \,e^{2} f \,g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-6 c^{2} d^{3} f \,g^{2} x \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-2 c^{2} d^{2} e \,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 )+4 a c d \,e^{2} f^{2} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-3 c^{2} d^{3} f^{2} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-c^{2} d^{2} e \,f^{3} \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-4 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,e^{2} g^{2} x +3 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c \,d^{2} g^{2} x +\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d e f g x -2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a d e \,g^{2}-2 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, a \,e^{2} f g +5 \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c \,d^{2} f g -\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d e \,f^{2}\right )}{4 \sqrt {e x +d}\, \sqrt {\left (a e g -c d f \right ) g}\, \left (g x +f \right )^{2} \left (a e g -c d f \right )^{2} \sqrt {c d x +a e}\, 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 {{\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 )}^{3}}\,{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 {{\left (d+e\,x\right )}^{3/2}}{{\left (f+g\,x\right )}^3\,\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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