Optimal. Leaf size=267 \[ \frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}} \]
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Rubi [A]
time = 0.21, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {32 c^3 d^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}+\frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}+\frac {12 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{35 \sqrt {d+e x} (f+g x)^{5/2} (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}}{7 \sqrt {d+e x} (f+g x)^{7/2} (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)^{9/2} \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}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {(6 c d) \int \frac {\sqrt {d+e x}}{(f+g x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 (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}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {\left (24 c^2 d^2\right ) \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}{35 (c d f-a e g)^2}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (16 c^3 d^3\right ) \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}{35 (c d f-a e g)^3}\\ &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 (c d f-a e g) \sqrt {d+e x} (f+g x)^{7/2}}+\frac {12 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{5/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^3 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {32 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 (c d f-a e g)^4 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 152, normalized size = 0.57 \begin {gather*} \frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-5 a^3 e^3 g^3+3 a^2 c d e^2 g^2 (7 f+2 g x)-a c^2 d^2 e g \left (35 f^2+28 f g x+8 g^2 x^2\right )+c^3 d^3 \left (35 f^3+70 f^2 g x+56 f g^2 x^2+16 g^3 x^3\right )\right )}{35 (c d f-a e g)^4 \sqrt {d+e x} (f+g x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 183, normalized size = 0.69
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right )}{35 \sqrt {e x +d}\, \left (g x +f \right )^{\frac {7}{2}} \left (a e g -c d f \right )^{4}}\) | \(183\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}-56 c^{3} d^{3} f \,g^{2} x^{2}-6 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -70 c^{3} d^{3} f^{2} g x +5 a^{3} e^{3} g^{3}-21 a^{2} c d \,e^{2} f \,g^{2}+35 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}{35 \left (g x +f \right )^{\frac {7}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(260\) |
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. 1002 vs.
\(2 (247) = 494\).
time = 4.10, size = 1002, normalized size = 3.75 \begin {gather*} \frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 56 \, c^{3} d^{3} f g^{2} x^{2} + 70 \, c^{3} d^{3} f^{2} g x + 35 \, c^{3} d^{3} f^{3} - 5 \, a^{3} g^{3} e^{3} + 3 \, {\left (2 \, a^{2} c d g^{3} x + 7 \, a^{2} c d f g^{2}\right )} e^{2} - {\left (8 \, a c^{2} d^{2} g^{3} x^{2} + 28 \, a c^{2} d^{2} f g^{2} x + 35 \, a c^{2} d^{2} f^{2} g\right )} 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}}{35 \, {\left (c^{4} d^{5} f^{4} g^{4} x^{4} + 4 \, c^{4} d^{5} f^{5} g^{3} x^{3} + 6 \, c^{4} d^{5} f^{6} g^{2} x^{2} + 4 \, c^{4} d^{5} f^{7} g x + c^{4} d^{5} f^{8} + {\left (a^{4} g^{8} x^{5} + 4 \, a^{4} f g^{7} x^{4} + 6 \, a^{4} f^{2} g^{6} x^{3} + 4 \, a^{4} f^{3} g^{5} x^{2} + a^{4} f^{4} g^{4} x\right )} e^{5} - {\left (4 \, a^{3} c d f g^{7} x^{5} - a^{4} d f^{4} g^{4} + {\left (16 \, a^{3} c d f^{2} g^{6} - a^{4} d g^{8}\right )} x^{4} + 4 \, {\left (6 \, a^{3} c d f^{3} g^{5} - a^{4} d f g^{7}\right )} x^{3} + 2 \, {\left (8 \, a^{3} c d f^{4} g^{4} - 3 \, a^{4} d f^{2} g^{6}\right )} x^{2} + 4 \, {\left (a^{3} c d f^{5} g^{3} - a^{4} d f^{3} g^{5}\right )} x\right )} e^{4} + 2 \, {\left (3 \, a^{2} c^{2} d^{2} f^{2} g^{6} x^{5} - 2 \, a^{3} c d^{2} f^{5} g^{3} + 2 \, {\left (6 \, a^{2} c^{2} d^{2} f^{3} g^{5} - a^{3} c d^{2} f g^{7}\right )} x^{4} + 2 \, {\left (9 \, a^{2} c^{2} d^{2} f^{4} g^{4} - 4 \, a^{3} c d^{2} f^{2} g^{6}\right )} x^{3} + 12 \, {\left (a^{2} c^{2} d^{2} f^{5} g^{3} - a^{3} c d^{2} f^{3} g^{5}\right )} x^{2} + {\left (3 \, a^{2} c^{2} d^{2} f^{6} g^{2} - 8 \, a^{3} c d^{2} f^{4} g^{4}\right )} x\right )} e^{3} - 2 \, {\left (2 \, a c^{3} d^{3} f^{3} g^{5} x^{5} - 3 \, a^{2} c^{2} d^{3} f^{6} g^{2} + {\left (8 \, a c^{3} d^{3} f^{4} g^{4} - 3 \, a^{2} c^{2} d^{3} f^{2} g^{6}\right )} x^{4} + 12 \, {\left (a c^{3} d^{3} f^{5} g^{3} - a^{2} c^{2} d^{3} f^{3} g^{5}\right )} x^{3} + 2 \, {\left (4 \, a c^{3} d^{3} f^{6} g^{2} - 9 \, a^{2} c^{2} d^{3} f^{4} g^{4}\right )} x^{2} + 2 \, {\left (a c^{3} d^{3} f^{7} g - 6 \, a^{2} c^{2} d^{3} f^{5} g^{3}\right )} x\right )} e^{2} + {\left (c^{4} d^{4} f^{4} g^{4} x^{5} - 4 \, a c^{3} d^{4} f^{7} g + 4 \, {\left (c^{4} d^{4} f^{5} g^{3} - a c^{3} d^{4} f^{3} g^{5}\right )} x^{4} + 2 \, {\left (3 \, c^{4} d^{4} f^{6} g^{2} - 8 \, a c^{3} d^{4} f^{4} g^{4}\right )} x^{3} + 4 \, {\left (c^{4} d^{4} f^{7} g - 6 \, a c^{3} d^{4} f^{5} g^{3}\right )} x^{2} + {\left (c^{4} d^{4} f^{8} - 16 \, a c^{3} d^{4} f^{6} 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(-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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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 = 5.51, size = 357, normalized size = 1.34 \begin {gather*} -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (10\,a^3\,e^3\,g^3-42\,a^2\,c\,d\,e^2\,f\,g^2+70\,a\,c^2\,d^2\,e\,f^2\,g-70\,c^3\,d^3\,f^3\right )}{35\,e\,g^3\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {32\,c^3\,d^3\,x^3\,\sqrt {d+e\,x}}{35\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}-\frac {4\,c\,d\,x\,\sqrt {d+e\,x}\,\left (3\,a^2\,e^2\,g^2-14\,a\,c\,d\,e\,f\,g+35\,c^2\,d^2\,f^2\right )}{35\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c^2\,d^2\,x^2\,\left (a\,e\,g-7\,c\,d\,f\right )\,\sqrt {d+e\,x}}{35\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {d\,f^3\,\sqrt {f+g\,x}}{e\,g^3}+\frac {x^3\,\sqrt {f+g\,x}\,\left (d\,g+3\,e\,f\right )}{e\,g}+\frac {3\,f\,x^2\,\sqrt {f+g\,x}\,\left (d\,g+e\,f\right )}{e\,g^2}+\frac {f^2\,x\,\sqrt {f+g\,x}\,\left (3\,d\,g+e\,f\right )}{e\,g^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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