Optimal. Leaf size=129 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g) \sqrt {d+e x} (f+g x)^{3/2}}+\frac {4 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 (c d f-a e g)^2 \sqrt {d+e x} \sqrt {f+g x}} \]
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Rubi [A]
time = 0.10, 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 = {886, 874}
\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.
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Rule 874
Rule 886
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*}
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Mathematica [A]
time = 0.12, size = 69, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} (-a e g+c d (3 f+2 g x))}{3 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 61, normalized size = 0.47
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-2 c d g x +a e g -3 c d f \right )}{3 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {3}{2}} \left (a e g -c d f \right )^{2}}\) | \(61\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-2 c d g x +a e g -3 c d f \right ) \sqrt {e x +d}}{3 \left (g x +f \right )^{\frac {3}{2}} \left (a^{2} e^{2} g^{2}-2 a c d e f g +f^{2} c^{2} d^{2}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(98\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 302 vs.
\(2 (119) = 238\).
time = 3.21, size = 302, normalized size = 2.34 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d g x + 3 \, c d f - a g e\right )} \sqrt {g x + f} \sqrt {x e + d}}{3 \, {\left (c^{2} d^{3} f^{2} g^{2} x^{2} + 2 \, c^{2} d^{3} f^{3} g x + c^{2} d^{3} f^{4} + {\left (a^{2} g^{4} x^{3} + 2 \, a^{2} f g^{3} x^{2} + a^{2} f^{2} g^{2} x\right )} e^{3} - {\left (2 \, a c d f g^{3} x^{3} - a^{2} d f^{2} g^{2} + {\left (4 \, a c d f^{2} g^{2} - a^{2} d g^{4}\right )} x^{2} + 2 \, {\left (a c d f^{3} g - a^{2} d f g^{3}\right )} x\right )} e^{2} + {\left (c^{2} d^{2} f^{2} g^{2} x^{3} - 2 \, a c d^{2} f^{3} g + 2 \, {\left (c^{2} d^{2} f^{3} g - a c d^{2} f g^{3}\right )} x^{2} + {\left (c^{2} d^{2} f^{4} - 4 \, a c d^{2} f^{2} g^{2}\right )} x\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 {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (f + g x\right )^{\frac {5}{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 [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.
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