Optimal. Leaf size=262 \[ -\frac {32 c^2 d^2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}-\frac {16 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac {12 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
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Rubi [A] time = 0.33, antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {868, 872, 860} \[ -\frac {32 c^2 d^2 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} \sqrt {f+g x} (c d f-a e g)^4}-\frac {16 c d g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{3/2} (c d f-a e g)^3}-\frac {12 g \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{5 \sqrt {d+e x} (f+g x)^{5/2} (c d f-a e g)^2}-\frac {2 \sqrt {d+e x}}{(f+g x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 860
Rule 868
Rule 872
Rubi steps
\begin {align*} \int \frac {(d+e x)^{3/2}}{(f+g x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {(6 g) \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}{c d f-a e g}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \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)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {(24 c d g) \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)^2}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \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)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \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)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {\left (16 c^2 d^2 g\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}{5 (c d f-a e g)^3}\\ &=-\frac {2 \sqrt {d+e x}}{(c d f-a e g) (f+g x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {12 g \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)^2 \sqrt {d+e x} (f+g x)^{5/2}}-\frac {16 c d g \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)^3 \sqrt {d+e x} (f+g x)^{3/2}}-\frac {32 c^2 d^2 g \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)^4 \sqrt {d+e x} \sqrt {f+g x}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 150, normalized size = 0.57 \[ -\frac {2 \sqrt {d+e x} \left (a^3 e^3 g^3-a^2 c d e^2 g^2 (5 f+2 g x)+a c^2 d^2 e g \left (15 f^2+20 f g x+8 g^2 x^2\right )+c^3 d^3 \left (5 f^3+30 f^2 g x+40 f g^2 x^2+16 g^3 x^3\right )\right )}{5 (f+g x)^{5/2} \sqrt {(d+e x) (a e+c d x)} (c d f-a e g)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.99, size = 1062, normalized size = 4.05 \[ -\frac {2 \, {\left (16 \, c^{3} d^{3} g^{3} x^{3} + 5 \, c^{3} d^{3} f^{3} + 15 \, a c^{2} d^{2} e f^{2} g - 5 \, a^{2} c d e^{2} f g^{2} + a^{3} e^{3} g^{3} + 8 \, {\left (5 \, c^{3} d^{3} f g^{2} + a c^{2} d^{2} e g^{3}\right )} x^{2} + 2 \, {\left (15 \, c^{3} d^{3} f^{2} g + 10 \, a c^{2} d^{2} e f g^{2} - a^{2} c d e^{2} g^{3}\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}}{5 \, {\left (a c^{4} d^{5} e f^{7} - 4 \, a^{2} c^{3} d^{4} e^{2} f^{6} g + 6 \, a^{3} c^{2} d^{3} e^{3} f^{5} g^{2} - 4 \, a^{4} c d^{2} e^{4} f^{4} g^{3} + a^{5} d e^{5} f^{3} g^{4} + {\left (c^{5} d^{5} e f^{4} g^{3} - 4 \, a c^{4} d^{4} e^{2} f^{3} g^{4} + 6 \, a^{2} c^{3} d^{3} e^{3} f^{2} g^{5} - 4 \, a^{3} c^{2} d^{2} e^{4} f g^{6} + a^{4} c d e^{5} g^{7}\right )} x^{5} + {\left (3 \, c^{5} d^{5} e f^{5} g^{2} + {\left (c^{5} d^{6} - 11 \, a c^{4} d^{4} e^{2}\right )} f^{4} g^{3} - 2 \, {\left (2 \, a c^{4} d^{5} e - 7 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{3} g^{4} + 6 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{2} g^{5} - {\left (4 \, a^{3} c^{2} d^{3} e^{3} + a^{4} c d e^{5}\right )} f g^{6} + {\left (a^{4} c d^{2} e^{4} + a^{5} e^{6}\right )} g^{7}\right )} x^{4} + {\left (3 \, c^{5} d^{5} e f^{6} g + a^{5} d e^{5} g^{7} + 3 \, {\left (c^{5} d^{6} - 3 \, a c^{4} d^{4} e^{2}\right )} f^{5} g^{2} - {\left (11 \, a c^{4} d^{5} e - 6 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{4} g^{3} + 2 \, {\left (7 \, a^{2} c^{3} d^{4} e^{2} + 3 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{3} g^{4} - 3 \, {\left (2 \, a^{3} c^{2} d^{3} e^{3} + 3 \, a^{4} c d e^{5}\right )} f^{2} g^{5} - {\left (a^{4} c d^{2} e^{4} - 3 \, a^{5} e^{6}\right )} f g^{6}\right )} x^{3} + {\left (c^{5} d^{5} e f^{7} + 3 \, a^{5} d e^{5} f g^{6} + {\left (3 \, c^{5} d^{6} - a c^{4} d^{4} e^{2}\right )} f^{6} g - 3 \, {\left (3 \, a c^{4} d^{5} e + 2 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{5} g^{2} + 2 \, {\left (3 \, a^{2} c^{3} d^{4} e^{2} + 7 \, a^{3} c^{2} d^{2} e^{4}\right )} f^{4} g^{3} + {\left (6 \, a^{3} c^{2} d^{3} e^{3} - 11 \, a^{4} c d e^{5}\right )} f^{3} g^{4} - 3 \, {\left (3 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f^{2} g^{5}\right )} x^{2} + {\left (3 \, a^{5} d e^{5} f^{2} g^{5} + {\left (c^{5} d^{6} + a c^{4} d^{4} e^{2}\right )} f^{7} - {\left (a c^{4} d^{5} e + 4 \, a^{2} c^{3} d^{3} e^{3}\right )} f^{6} g - 6 \, {\left (a^{2} c^{3} d^{4} e^{2} - a^{3} c^{2} d^{2} e^{4}\right )} f^{5} g^{2} + 2 \, {\left (7 \, a^{3} c^{2} d^{3} e^{3} - 2 \, a^{4} c d e^{5}\right )} f^{4} g^{3} - {\left (11 \, a^{4} c d^{2} e^{4} - a^{5} e^{6}\right )} f^{3} g^{4}\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 = 259, normalized size = 0.99 \[ -\frac {2 \left (c d x +a e \right ) \left (16 g^{3} x^{3} c^{3} d^{3}+8 a \,c^{2} d^{2} e \,g^{3} x^{2}+40 c^{3} d^{3} f \,g^{2} x^{2}-2 a^{2} c d \,e^{2} g^{3} x +20 a \,c^{2} d^{2} e f \,g^{2} x +30 c^{3} d^{3} f^{2} g x +a^{3} e^{3} g^{3}-5 a^{2} c d \,e^{2} f \,g^{2}+15 a \,c^{2} d^{2} e \,f^{2} g +5 f^{3} c^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{5 \left (g x +f \right )^{\frac {5}{2}} \left (g^{4} e^{4} a^{4}-4 a^{3} c d \,e^{3} f \,g^{3}+6 a^{2} c^{2} d^{2} e^{2} f^{2} g^{2}-4 a \,c^{3} d^{3} e \,f^{3} g +f^{4} c^{4} d^{4}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.70, size = 414, normalized size = 1.58 \[ -\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {4\,x\,\sqrt {d+e\,x}\,\left (-a^2\,e^2\,g^2+10\,a\,c\,d\,e\,f\,g+15\,c^2\,d^2\,f^2\right )}{5\,e\,g\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {\sqrt {d+e\,x}\,\left (\frac {2\,a^3\,e^3\,g^3}{5}-2\,a^2\,c\,d\,e^2\,f\,g^2+6\,a\,c^2\,d^2\,e\,f^2\,g+2\,c^3\,d^3\,f^3\right )}{c\,d\,e\,g^2\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {32\,c^2\,d^2\,g\,x^3\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}+\frac {16\,c\,d\,x^2\,\left (a\,e\,g+5\,c\,d\,f\right )\,\sqrt {d+e\,x}}{5\,e\,{\left (a\,e\,g-c\,d\,f\right )}^4}\right )}{x^4\,\sqrt {f+g\,x}+\frac {a\,f^2\,\sqrt {f+g\,x}}{c\,g^2}+\frac {x^2\,\sqrt {f+g\,x}\,\left (2\,c\,d^2\,f\,g+c\,d\,e\,f^2+a\,d\,e\,g^2+2\,a\,e^2\,f\,g\right )}{c\,d\,e\,g^2}+\frac {x^3\,\sqrt {f+g\,x}\,\left (c\,g\,d^2+2\,c\,f\,d\,e+a\,g\,e^2\right )}{c\,d\,e\,g}+\frac {f\,x\,\sqrt {f+g\,x}\,\left (c\,f\,d^2+2\,a\,g\,d\,e+a\,f\,e^2\right )}{c\,d\,e\,g^2}} \]
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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