Optimal. Leaf size=129 \[ \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \begin {gather*} \frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^{3/2} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{7/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {(2 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x} (f+g x)^{5/2}} \, dx}{5 (c d f-a e g)}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{5 (c d f-a e g) (d+e x)^{3/2} (f+g x)^{5/2}}+\frac {4 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{15 (c d f-a e g)^2 (d+e x)^{3/2} (f+g x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 0.53 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2} (c d (5 f+2 g x)-3 a e g)}{15 (d+e x)^{3/2} (f+g x)^{5/2} (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.13, size = 196, normalized size = 1.52 \begin {gather*} \frac {2 \sqrt {d+e x} (e f+e g x)^{7/2} \sqrt {a e^2+c d e x} \left (\frac {5 c d \left (a e^2-c d^2+c d (d+e x)\right )^{3/2}}{(g (d+e x)-d g+e f)^{3/2}}-\frac {3 g \left (a e^2-c d^2+c d (d+e x)\right )^{5/2}}{(g (d+e x)-d g+e f)^{5/2}}\right )}{15 e^4 \sqrt {\frac {(d+e x) \left (a e^2+c d e x\right )}{e}} \left (\frac {g (d+e x)-d g+e f}{e}\right )^{7/2} (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 402, normalized size = 3.12 \begin {gather*} \frac {2 \, {\left (2 \, c^{2} d^{2} g x^{2} + 5 \, a c d e f - 3 \, a^{2} e^{2} g + {\left (5 \, c^{2} d^{2} f - a c d e g\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{15 \, {\left (c^{2} d^{3} f^{5} - 2 \, a c d^{2} e f^{4} g + a^{2} d e^{2} f^{3} g^{2} + {\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^{4} + {\left (3 \, c^{2} d^{2} e f^{3} g^{2} + a^{2} d e^{2} g^{5} + {\left (c^{2} d^{3} - 6 \, a c d e^{2}\right )} f^{2} g^{3} - {\left (2 \, a c d^{2} e - 3 \, a^{2} e^{3}\right )} f g^{4}\right )} x^{3} + 3 \, {\left (c^{2} d^{2} e f^{4} g + a^{2} d e^{2} f g^{4} + {\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{3} g^{2} - {\left (2 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{5} + 3 \, a^{2} d e^{2} f^{2} g^{3} + {\left (3 \, c^{2} d^{3} - 2 \, a c d e^{2}\right )} f^{4} g - {\left (6 \, a c d^{2} e - a^{2} e^{3}\right )} f^{3} g^{2}\right )} x\right )}} \end {gather*}
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 [A] time = 0.01, size = 99, normalized size = 0.77 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +3 a e g -5 c d f \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}} \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 {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d} {\left (g x + f\right )}^{\frac {7}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 187, normalized size = 1.45 \begin {gather*} \frac {\left (\frac {x\,\left (10\,c^2\,d^2\,f-2\,a\,c\,d\,e\,g\right )}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {6\,a^2\,e^2\,g-10\,a\,c\,d\,e\,f}{15\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^2}+\frac {4\,c^2\,d^2\,x^2}{15\,g\,{\left (a\,e\,g-c\,d\,f\right )}^2}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}+\frac {f^2\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g^2}+\frac {2\,f\,x\,\sqrt {f+g\,x}\,\sqrt {d+e\,x}}{g}} \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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