Optimal. Leaf size=161 \[ -\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \sqrt {g} \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {880, 905, 65,
223, 212} \begin {gather*} \frac {2 \sqrt {g} \sqrt {d+e x} \sqrt {a e+c d x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 223
Rule 880
Rule 905
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2} \sqrt {f+g x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {g \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d}\\ &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (g \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 g \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (2 g \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ &=-\frac {2 \sqrt {d+e x} \sqrt {f+g x}}{c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {2 \sqrt {g} \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{3/2} d^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 117, normalized size = 0.73 \begin {gather*} -\frac {2 \sqrt {d+e x} \left (\sqrt {c} \sqrt {d} \sqrt {f+g x}-\sqrt {g} \sqrt {a e+c d x} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{c^{3/2} d^{3/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 200, normalized size = 1.24
method | result | size |
default | \(\frac {\sqrt {g x +f}\, \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) c d g x +\ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}}{2 \sqrt {d g c}}\right ) a e g -2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {d g c}\right )}{\sqrt {d g c}\, \left (c d x +a e \right ) \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, d c \sqrt {e x +d}}\) | \(200\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.83, size = 577, normalized size = 3.58 \begin {gather*} \left [\frac {{\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )} \sqrt {\frac {g}{c d}} \log \left (-\frac {8 \, c^{2} d^{3} g^{2} x^{2} + 8 \, c^{2} d^{3} f g x + c^{2} d^{3} f^{2} + a^{2} g^{2} x e^{3} + 4 \, {\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d g e\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d} \sqrt {\frac {g}{c d}} + {\left (8 \, a c d g^{2} x^{2} + 6 \, a c d f g x + a^{2} d g^{2}\right )} e^{2} + {\left (8 \, c^{2} d^{2} g^{2} x^{3} + 8 \, c^{2} d^{2} f g x^{2} + 6 \, a c d^{2} f g + {\left (c^{2} d^{2} f^{2} + 8 \, a c d^{2} g^{2}\right )} x\right )} e}{x e + d}\right ) - 4 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{2 \, {\left (c^{2} d^{3} x + a c d x e^{2} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e\right )}}, -\frac {{\left (c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e\right )} \sqrt {-\frac {g}{c d}} \arctan \left (\frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d} c d \sqrt {-\frac {g}{c d}}}{2 \, c d^{2} g x + c d^{2} f + a g x e^{2} + {\left (2 \, c d g x^{2} + c d f x + a d g\right )} e}\right ) + 2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {g x + f} \sqrt {x e + d}}{c^{2} d^{3} x + a c d x e^{2} + {\left (c^{2} d^{2} x^{2} + a c d^{2}\right )} e}\right ] \end {gather*}
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 \begin {gather*} \int \frac {\left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {f+g\,x}\,{\left (d+e\,x\right )}^{3/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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