Integrand size = 46, antiderivative size = 269 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {16 (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g (c d f-a e g)^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3 e}+\frac {12 (c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt {d+e x}}+\frac {2 (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt {d+e x}} \]
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Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {884, 808, 662} \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {16 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right )}{35 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g \sqrt {d+e x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)^2}{35 c^3 d^3 e}+\frac {12 (f+g x)^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}{35 c^2 d^2 \sqrt {d+e x}}+\frac {2 (f+g x)^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{7 c d \sqrt {d+e x}} \]
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Rule 662
Rule 808
Rule 884
Rubi steps \begin{align*} \text {integral}& = \frac {2 (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt {d+e x}}+\frac {\left (6 \left (c d e^2 f+c d^2 e g-e \left (c d^2+a e^2\right ) g\right )\right ) \int \frac {\sqrt {d+e x} (f+g x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{7 c d e^2} \\ & = \frac {12 (c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt {d+e x}}+\frac {2 (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt {d+e x}}+\frac {\left (24 (c d f-a e g)^2\right ) \int \frac {\sqrt {d+e x} (f+g x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 c^2 d^2} \\ & = \frac {16 g (c d f-a e g)^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3 e}+\frac {12 (c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt {d+e x}}+\frac {2 (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt {d+e x}}-\frac {\left (8 (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{35 c^3 d^3 e} \\ & = -\frac {16 (c d f-a e g)^2 \left (2 a e^2 g-c d (3 e f-d g)\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^4 d^4 e \sqrt {d+e x}}+\frac {16 g (c d f-a e g)^2 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^3 d^3 e}+\frac {12 (c d f-a e g) (f+g x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{35 c^2 d^2 \sqrt {d+e x}}+\frac {2 (f+g x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{7 c d \sqrt {d+e x}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (-16 a^3 e^3 g^3+8 a^2 c d e^2 g^2 (7 f+g x)-2 a c^2 d^2 e g \left (35 f^2+14 f g x+3 g^2 x^2\right )+c^3 d^3 \left (35 f^3+35 f^2 g x+21 f g^2 x^2+5 g^3 x^3\right )\right )}{35 c^4 d^4 \sqrt {d+e x}} \]
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Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {2 \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (-5 g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-21 c^{3} d^{3} f \,g^{2} x^{2}-8 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -35 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-56 a^{2} c d \,e^{2} f \,g^{2}+70 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right )}{35 \sqrt {e x +d}\, c^{4} d^{4}}\) | \(170\) |
gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-5 g^{3} x^{3} c^{3} d^{3}+6 a \,c^{2} d^{2} e \,g^{3} x^{2}-21 c^{3} d^{3} f \,g^{2} x^{2}-8 a^{2} c d \,e^{2} g^{3} x +28 a \,c^{2} d^{2} e f \,g^{2} x -35 c^{3} d^{3} f^{2} g x +16 a^{3} e^{3} g^{3}-56 a^{2} c d \,e^{2} f \,g^{2}+70 a \,c^{2} d^{2} e \,f^{2} g -35 f^{3} c^{3} d^{3}\right ) \sqrt {e x +d}}{35 c^{4} d^{4} \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}\) | \(188\) |
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Time = 0.30 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, {\left (5 \, c^{3} d^{3} g^{3} x^{3} + 35 \, c^{3} d^{3} f^{3} - 70 \, a c^{2} d^{2} e f^{2} g + 56 \, a^{2} c d e^{2} f g^{2} - 16 \, a^{3} e^{3} g^{3} + 3 \, {\left (7 \, c^{3} d^{3} f g^{2} - 2 \, a c^{2} d^{2} e g^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{3} f^{2} g - 28 \, a c^{2} d^{2} e f g^{2} + 8 \, a^{2} c d e^{2} g^{3}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{35 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
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\[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\sqrt {d + e x} \left (f + g x\right )^{3}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
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Time = 0.24 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, \sqrt {c d x + a e} f^{3}}{c d} + \frac {2 \, {\left (c^{2} d^{2} x^{2} - a c d e x - 2 \, a^{2} e^{2}\right )} f^{2} g}{\sqrt {c d x + a e} c^{2} d^{2}} + \frac {2 \, {\left (3 \, c^{3} d^{3} x^{3} - a c^{2} d^{2} e x^{2} + 4 \, a^{2} c d e^{2} x + 8 \, a^{3} e^{3}\right )} f g^{2}}{5 \, \sqrt {c d x + a e} c^{3} d^{3}} + \frac {2 \, {\left (5 \, c^{4} d^{4} x^{4} - a c^{3} d^{3} e x^{3} + 2 \, a^{2} c^{2} d^{2} e^{2} x^{2} - 8 \, a^{3} c d e^{3} x - 16 \, a^{4} e^{4}\right )} g^{3}}{35 \, \sqrt {c d x + a e} c^{4} d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (245) = 490\).
Time = 0.31 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.28 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \, e {\left (\frac {35 \, {\left (c^{3} d^{3} f^{3} - 3 \, a c^{2} d^{2} e f^{2} g + 3 \, a^{2} c d e^{2} f g^{2} - a^{3} e^{3} g^{3}\right )} \sqrt {{\left (e x + d\right )} c d e - c d^{2} e + a e^{3}}}{c^{4} d^{4} e} - \frac {35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{3} e^{3} f^{3} - 35 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{4} e^{2} f^{2} g - 70 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{2} e^{4} f^{2} g + 21 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{5} e f g^{2} + 28 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{3} e^{3} f g^{2} + 56 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d e^{5} f g^{2} - 5 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} g^{3} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} g^{3} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} g^{3} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6} g^{3}}{c^{4} d^{4} e^{4}} + \frac {35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{4} f^{2} g - 70 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c d e^{5} f g^{2} + 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} g^{3} + 21 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c d e^{2} f g^{2} - 21 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} g^{3} + 5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} g^{3}}{c^{4} d^{4} e^{7}}\right )}}{35 \, {\left | e \right |}} \]
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Time = 12.61 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {d+e x} (f+g x)^3}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {\sqrt {d+e\,x}\,\left (32\,a^3\,e^3\,g^3-112\,a^2\,c\,d\,e^2\,f\,g^2+140\,a\,c^2\,d^2\,e\,f^2\,g-70\,c^3\,d^3\,f^3\right )}{35\,c^4\,d^4\,e}-\frac {2\,g^3\,x^3\,\sqrt {d+e\,x}}{7\,c\,d\,e}+\frac {6\,g^2\,x^2\,\left (2\,a\,e\,g-7\,c\,d\,f\right )\,\sqrt {d+e\,x}}{35\,c^2\,d^2\,e}-\frac {2\,g\,x\,\sqrt {d+e\,x}\,\left (8\,a^2\,e^2\,g^2-28\,a\,c\,d\,e\,f\,g+35\,c^2\,d^2\,f^2\right )}{35\,c^3\,d^3\,e}\right )}{x+\frac {d}{e}} \]
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