Optimal. Leaf size=198 \[ \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)} \]
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Rubi [A] time = 0.22, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {872, 860} \[ \frac {16 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^3}+\frac {8 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{15 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^2}+\frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 872
Rubi steps
\begin {align*} \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 &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {(4 c d) \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}{5 (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}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\left (8 c^2 d^2\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}{15 (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}}{5 (c d f-a e g) \sqrt {d+e x} (f+g x)^{5/2}}+\frac {8 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^2 \sqrt {d+e x} (f+g x)^{3/2}}+\frac {16 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{15 (c d f-a e g)^3 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 0.53 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (3 a^2 e^2 g^2-2 a c d e g (5 f+2 g x)+c^2 d^2 \left (15 f^2+20 f g x+8 g^2 x^2\right )\right )}{15 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 572, normalized size = 2.89 \[ \frac {2 \, {\left (8 \, c^{2} d^{2} g^{2} x^{2} + 15 \, c^{2} d^{2} f^{2} - 10 \, a c d e f g + 3 \, a^{2} e^{2} g^{2} + 4 \, {\left (5 \, c^{2} d^{2} f g - a c d e g^{2}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d} \sqrt {g x + f}}{15 \, {\left (c^{3} d^{4} f^{6} - 3 \, a c^{2} d^{3} e f^{5} g + 3 \, a^{2} c d^{2} e^{2} f^{4} g^{2} - a^{3} d e^{3} f^{3} g^{3} + {\left (c^{3} d^{3} e f^{3} g^{3} - 3 \, a c^{2} d^{2} e^{2} f^{2} g^{4} + 3 \, a^{2} c d e^{3} f g^{5} - a^{3} e^{4} g^{6}\right )} x^{4} + {\left (3 \, c^{3} d^{3} e f^{4} g^{2} - a^{3} d e^{3} g^{6} + {\left (c^{3} d^{4} - 9 \, a c^{2} d^{2} e^{2}\right )} f^{3} g^{3} - 3 \, {\left (a c^{2} d^{3} e - 3 \, a^{2} c d e^{3}\right )} f^{2} g^{4} + 3 \, {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f g^{5}\right )} x^{3} + 3 \, {\left (c^{3} d^{3} e f^{5} g - a^{3} d e^{3} f g^{5} + {\left (c^{3} d^{4} - 3 \, a c^{2} d^{2} e^{2}\right )} f^{4} g^{2} - 3 \, {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{3} g^{3} + {\left (3 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{2} g^{4}\right )} x^{2} + {\left (c^{3} d^{3} e f^{6} - 3 \, a^{3} d e^{3} f^{2} g^{4} + 3 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} f^{5} g - 3 \, {\left (3 \, a c^{2} d^{3} e - a^{2} c d e^{3}\right )} f^{4} g^{2} + {\left (9 \, a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} f^{3} g^{3}\right )} x\right )}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 0.85 \[ -\frac {2 \left (c d x +a e \right ) \left (8 g^{2} x^{2} c^{2} d^{2}-4 a c d e \,g^{2} x +20 c^{2} d^{2} f g x +3 a^{2} e^{2} g^{2}-10 a c d e f g +15 f^{2} c^{2} d^{2}\right ) \sqrt {e x +d}}{15 \left (g x +f \right )^{\frac {5}{2}} \left (a^{3} e^{3} g^{3}-3 a^{2} c d \,e^{2} f \,g^{2}+3 a \,c^{2} d^{2} e \,f^{2} g -f^{3} c^{3} d^{3}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}} \]
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 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (g x + f\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.17, size = 242, normalized size = 1.22 \[ -\frac {\left (\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,e^2\,g^2-20\,a\,c\,d\,e\,f\,g+30\,c^2\,d^2\,f^2\right )}{15\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^3}+\frac {16\,c^2\,d^2\,x^2\,\sqrt {d+e\,x}}{15\,e\,{\left (a\,e\,g-c\,d\,f\right )}^3}-\frac {8\,c\,d\,x\,\left (a\,e\,g-5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{15\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^3}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x^3\,\sqrt {f+g\,x}+\frac {d\,f^2\,\sqrt {f+g\,x}}{e\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (d\,g+2\,e\,f\right )}{e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (2\,d\,g+e\,f\right )}{e\,g^2}} \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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