Integrand size = 48, antiderivative size = 289 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=-\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {5 g^{3/2} (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{7/2} d^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {880, 884, 905, 65, 223, 212} \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {5 g^{3/2} \sqrt {d+e x} \sqrt {a e+c d x} (c d f-a e g) \text {arctanh}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{7/2} d^{7/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
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Rule 65
Rule 212
Rule 223
Rule 880
Rule 884
Rule 905
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(5 g) \int \frac {(d+e x)^{3/2} (f+g x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (5 g^2\right ) \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c^2 d^2} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {\left (5 g^2 (c d f-a e g)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^3 d^3} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {\left (5 g^2 (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \int \frac {1}{\sqrt {a e+c d x} \sqrt {f+g x}} \, dx}{2 c^3 d^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {\left (5 g^2 (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {f-\frac {a e g}{c d}+\frac {g x^2}{c d}}} \, dx,x,\sqrt {a e+c d x}\right )}{c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {\left (5 g^2 (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {g x^2}{c d}} \, dx,x,\frac {\sqrt {a e+c d x}}{\sqrt {f+g x}}\right )}{c^4 d^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = -\frac {2 (d+e x)^{3/2} (f+g x)^{5/2}}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {10 g \sqrt {d+e x} (f+g x)^{3/2}}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {5 g^2 \sqrt {f+g x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^3 d^3 \sqrt {d+e x}}+\frac {5 g^{3/2} (c d f-a e g) \sqrt {a e+c d x} \sqrt {d+e x} \tanh ^{-1}\left (\frac {\sqrt {g} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {f+g x}}\right )}{c^{7/2} d^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.65 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {(d+e x)^{5/2} \left (\sqrt {c} \sqrt {d} (a e+c d x) \sqrt {f+g x} \left (15 a^2 e^2 g^2-10 a c d e g (f-2 g x)+c^2 d^2 \left (-2 f^2-14 f g x+3 g^2 x^2\right )\right )+15 g^{3/2} (c d f-a e g) (a e+c d x)^{5/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {f+g x}}{\sqrt {g} \sqrt {a e+c d x}}\right )\right )}{3 c^{7/2} d^{7/2} ((a e+c d x) (d+e x))^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(641\) vs. \(2(245)=490\).
Time = 0.55 (sec) , antiderivative size = 642, normalized size of antiderivative = 2.22
method | result | size |
default | \(-\frac {\left (15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e \,g^{3} x^{2}-15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) c^{3} d^{3} f \,g^{2} x^{2}+30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} g^{3} x -30 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a \,c^{2} d^{2} e f \,g^{2} x +15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{3} e^{3} g^{3}-15 \ln \left (\frac {2 c d g x +a e g +c d f +2 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}}{2 \sqrt {c d g}}\right ) a^{2} c d \,e^{2} f \,g^{2}-6 c^{2} d^{2} g^{2} x^{2} \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}-40 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, a c d e \,g^{2} x +28 \sqrt {c d g}\, \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, c^{2} d^{2} f g x -30 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a^{2} e^{2} g^{2}+20 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, a c d e f g +4 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, c^{2} d^{2} f^{2}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \sqrt {g x +f}}{6 \sqrt {\left (g x +f \right ) \left (c d x +a e \right )}\, \sqrt {c d g}\, \left (c d x +a e \right )^{2} c^{3} d^{3} \sqrt {e x +d}}\) | \(642\) |
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Time = 0.89 (sec) , antiderivative size = 1055, normalized size of antiderivative = 3.65 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\left [\frac {4 \, {\left (3 \, c^{2} d^{2} g^{2} x^{2} - 2 \, c^{2} d^{2} f^{2} - 10 \, a c d e f g + 15 \, a^{2} e^{2} g^{2} - 2 \, {\left (7 \, c^{2} d^{2} f g - 10 \, 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 (a^{2} c d^{2} e^{2} f g - a^{3} d e^{3} g^{2} + {\left (c^{3} d^{3} e f g - a c^{2} d^{2} e^{2} g^{2}\right )} x^{3} + {\left ({\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f g - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g - {\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{2}\right )} x\right )} \sqrt {\frac {g}{c d}} \log \left (-\frac {8 \, c^{2} d^{2} e g^{2} x^{3} + c^{2} d^{3} f^{2} + 6 \, a c d^{2} e f g + a^{2} d e^{2} g^{2} + 8 \, {\left (c^{2} d^{2} e f g + {\left (c^{2} d^{3} + a c d e^{2}\right )} g^{2}\right )} x^{2} - 4 \, {\left (2 \, c^{2} d^{2} g x + c^{2} d^{2} f + a c d e g\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} \sqrt {\frac {g}{c d}} + {\left (c^{2} d^{2} e f^{2} + 2 \, {\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} f g + {\left (8 \, a c d^{2} e + a^{2} e^{3}\right )} g^{2}\right )} x}{e x + d}\right )}{12 \, {\left (c^{5} d^{5} e x^{3} + a^{2} c^{3} d^{4} e^{2} + {\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} x^{2} + {\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} x\right )}}, \frac {2 \, {\left (3 \, c^{2} d^{2} g^{2} x^{2} - 2 \, c^{2} d^{2} f^{2} - 10 \, a c d e f g + 15 \, a^{2} e^{2} g^{2} - 2 \, {\left (7 \, c^{2} d^{2} f g - 10 \, 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 (a^{2} c d^{2} e^{2} f g - a^{3} d e^{3} g^{2} + {\left (c^{3} d^{3} e f g - a c^{2} d^{2} e^{2} g^{2}\right )} x^{3} + {\left ({\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2}\right )} f g - {\left (a c^{2} d^{3} e + 2 \, a^{2} c d e^{3}\right )} g^{2}\right )} x^{2} + {\left ({\left (2 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} f g - {\left (2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} g^{2}\right )} x\right )} \sqrt {-\frac {g}{c d}} \arctan \left (\frac {2 \, \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} c d \sqrt {-\frac {g}{c d}}}{2 \, c d e g x^{2} + c d^{2} f + a d e g + {\left (c d e f + {\left (2 \, c d^{2} + a e^{2}\right )} g\right )} x}\right )}{6 \, {\left (c^{5} d^{5} e x^{3} + a^{2} c^{3} d^{4} e^{2} + {\left (c^{5} d^{6} + 2 \, a c^{4} d^{4} e^{2}\right )} x^{2} + {\left (2 \, a c^{4} d^{5} e + a^{2} c^{3} d^{3} e^{3}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}} {\left (g x + f\right )}^{\frac {5}{2}}}{{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1105 vs. \(2 (245) = 490\).
Time = 0.86 (sec) , antiderivative size = 1105, normalized size of antiderivative = 3.82 \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\frac {15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} c^{2} d^{3} e f g^{3} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - 15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c d e^{3} f g^{3} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - 15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a c d^{2} e^{2} g^{4} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) + 15 \, \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} a^{2} e^{4} g^{4} \log \left ({\left | -\sqrt {e^{2} f - d e g} \sqrt {c d g} + \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \right |}\right ) - 2 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{2} e^{2} f^{2} g^{2} + 14 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{3} e f g^{3} - 10 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d e^{3} f g^{3} + 3 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} c^{2} d^{4} g^{4} - 20 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a c d^{2} e^{2} g^{4} + 15 \, \sqrt {e^{2} f - d e g} \sqrt {c d g} a^{2} e^{4} g^{4}}{3 \, {\left (\sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} c^{4} d^{5} e {\left | g \right |} - \sqrt {-c d^{2} e g^{2} + a e^{3} g^{2}} \sqrt {c d g} a c^{3} d^{3} e^{3} {\left | g \right |}\right )}} - \frac {\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} {\left ({\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} {\left (\frac {3 \, {\left (c^{5} d^{5} e^{2} f g^{5} - a c^{4} d^{4} e^{3} g^{6}\right )} {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}} - \frac {20 \, {\left (c^{5} d^{5} e^{4} f^{2} g^{5} - 2 \, a c^{4} d^{4} e^{5} f g^{6} + a^{2} c^{3} d^{3} e^{6} g^{7}\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}}\right )} + \frac {15 \, {\left (c^{5} d^{5} e^{6} f^{3} g^{5} - 3 \, a c^{4} d^{4} e^{7} f^{2} g^{6} + 3 \, a^{2} c^{3} d^{3} e^{8} f g^{7} - a^{3} c^{2} d^{2} e^{9} g^{8}\right )}}{c^{6} d^{6} e^{4} f g {\left | g \right |} - a c^{5} d^{5} e^{5} g^{2} {\left | g \right |}}\right )}}{3 \, {\left (c d e^{2} f g - a e^{3} g^{2} - {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g\right )} \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g}} - \frac {5 \, {\left (c d f g^{3} - a e g^{4}\right )} \log \left ({\left | -\sqrt {e^{2} f + {\left (e x + d\right )} e g - d e g} \sqrt {c d g} + \sqrt {-c d e^{2} f g + a e^{3} g^{2} + {\left (e^{2} f + {\left (e x + d\right )} e g - d e g\right )} c d g} \right |}\right )}{\sqrt {c d g} c^{3} d^{3} {\left | g \right |}} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2} (f+g x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^{5/2}}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \]
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