Optimal. Leaf size=129 \[ \frac {4 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)} \]
________________________________________________________________________________________
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 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 860
Rule 872
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^{3/2}}+\frac {(2 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 (c d f-a e g)}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^{3/2}}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 69, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(d+e x) (a e+c d x)} (c d (3 f+2 g x)-a e g)}{3 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 3.44, size = 196, normalized size = 1.52 \begin {gather*} \frac {2 \sqrt {d+e x} (e f+e g x)^{5/2} \sqrt {a e^2+c d e x} \left (\frac {3 c d \sqrt {a e^2-c d^2+c d (d+e x)}}{\sqrt {g (d+e x)-d g+e f}}-\frac {g \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}}\right )}{3 e^3 \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 )^{5/2} (c d f-a e g)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.42, size = 288, normalized size = 2.23 \begin {gather*} \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d g x + 3 \, c d f - a e g\right )} \sqrt {e x + d} \sqrt {g x + f}}{3 \, {\left (c^{2} d^{3} f^{4} - 2 \, a c d^{2} e f^{3} g + a^{2} d e^{2} f^{2} g^{2} + {\left (c^{2} d^{2} e f^{2} g^{2} - 2 \, a c d e^{2} f g^{3} + a^{2} e^{3} g^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{2} e f^{3} g + a^{2} d e^{2} g^{4} + {\left (c^{2} d^{3} - 4 \, a c d e^{2}\right )} f^{2} g^{2} - 2 \, {\left (a c d^{2} e - a^{2} e^{3}\right )} f g^{3}\right )} x^{2} + {\left (c^{2} d^{2} e f^{4} + 2 \, a^{2} d e^{2} f g^{3} + 2 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} f^{3} g - {\left (4 \, a c d^{2} e - a^{2} e^{3}\right )} f^{2} g^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 98, normalized size = 0.76 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +a e g -3 c d f \right ) \sqrt {e x +d}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{\frac {5}{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.90, size = 147, normalized size = 1.14 \begin {gather*} -\frac {\left (\frac {\left (2\,a\,e\,g-6\,c\,d\,f\right )\,\sqrt {d+e\,x}}{3\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^2}-\frac {4\,c\,d\,x\,\sqrt {d+e\,x}}{3\,e\,{\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}+\frac {d\,f\,\sqrt {f+g\,x}}{e\,g}+\frac {x\,\sqrt {f+g\,x}\,\left (d\,g+e\,f\right )}{e\,g}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________