Optimal. Leaf size=139 \[ \frac {2 e \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} \sqrt {c d f-a e g}} \]
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Rubi [A] time = 0.19, antiderivative size = 139, 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 = {880, 874, 205} \[ \frac {2 e \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d f-a e g}}\right )}{g^{3/2} \sqrt {c d f-a e g}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 874
Rule 880
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {\left (2 \left (\frac {1}{2} c d e^2 f-\frac {1}{2} c d^2 e 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}{c d e g}\\ &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {\left (2 e^2 (e f-d g)\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}\\ &=\frac {2 e \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt {d+e x}}-\frac {2 (e f-d g) \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} \sqrt {c d f-a e g}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 140, normalized size = 1.01 \[ \frac {2 \sqrt {d+e x} \left (e \sqrt {g} (a e+c d x) \sqrt {c d f-a e g}+c d (d g-e f) \sqrt {a e+c d x} \tan ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c d f-a e g}}\right )\right )}{c d g^{3/2} \sqrt {(d+e x) (a e+c d x)} \sqrt {c d f-a e g}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 511, normalized size = 3.68 \[ \left [\frac {{\left (c d^{2} e f - c d^{3} g + {\left (c d e^{2} f - c d^{2} e g\right )} x\right )} \sqrt {-c d f g + a e g^{2}} \log \left (-\frac {c d e g x^{2} - c d^{2} f + 2 \, a d e g - {\left (c d e f - {\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {-c d f g + a e g^{2}} \sqrt {e x + d}}{e g x^{2} + d f + {\left (e f + d g\right )} x}\right ) + 2 \, {\left (c d e f g - a e^{2} g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c^{2} d^{3} f g^{2} - a c d^{2} e g^{3} + {\left (c^{2} d^{2} e f g^{2} - a c d e^{2} g^{3}\right )} x}, \frac {2 \, {\left ({\left (c d^{2} e f - c d^{3} g + {\left (c d e^{2} f - c d^{2} e g\right )} x\right )} \sqrt {c d f g - a e g^{2}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {c d f g - a e g^{2}} \sqrt {e x + d}}{c d e g x^{2} + a d e g + {\left (c d^{2} + a e^{2}\right )} g x}\right ) + {\left (c d e f g - a e^{2} g^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}\right )}}{c^{2} d^{3} f g^{2} - a c d^{2} e g^{3} + {\left (c^{2} d^{2} e f g^{2} - a c d e^{2} g^{3}\right )} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 163, normalized size = 1.17 \[ -\frac {2 \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, \left (c \,d^{2} g \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-c d e f \arctanh \left (\frac {\sqrt {c d x +a e}\, g}{\sqrt {\left (a e g -c d f \right ) g}}\right )-\sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, e \right )}{\sqrt {e x +d}\, \sqrt {c d x +a e}\, \sqrt {\left (a e g -c d f \right ) g}\, c d g} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x\right )}^{3/2}}{\left (f+g\,x\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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