Integrand size = 27, antiderivative size = 325 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \]
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Time = 0.21 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {832, 793, 635, 212} \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {3 e \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (B \left (-4 c e (a e+3 b d)+5 b^2 e^2+8 c^2 d^2\right )+4 A c e (2 c d-b e)\right )}{8 c^{7/2}}+\frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \sqrt {a+b x+c x^2} \left (-2 c e x \left (-4 c (3 a B e+A b e+b B d)+8 A c^2 d+5 b^2 B e\right )+4 b c \left (-13 a B e^2+6 A c d e+4 B c d^2\right )-32 c^2 \left (-a A e^2-3 a B d e+A c d^2\right )-12 b^2 c e (A e+3 B d)+15 b^3 B e^2\right )}{4 c^3 \left (b^2-4 a c\right )} \]
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Rule 212
Rule 635
Rule 793
Rule 832
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}+\frac {2 \int \frac {(d+e x) \left (\frac {1}{2} e \left (b^2 B d+4 A b c d-12 a B c d+4 a b B e-8 a A c e\right )+\frac {1}{2} e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{c \left (b^2-4 a c\right )} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{8 c^3} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {\left (3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{4 c^3} \\ & = \frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {e \left (15 b^3 B e^2-12 b^2 c e (3 B d+A e)-32 c^2 \left (A c d^2-3 a B d e-a A e^2\right )+4 b c \left (4 B c d^2+6 A c d e-13 a B e^2\right )-2 c e \left (8 A c^2 d+5 b^2 B e-4 c (b B d+A b e+3 a B e)\right ) x\right ) \sqrt {a+b x+c x^2}}{4 c^3 \left (b^2-4 a c\right )}+\frac {3 e \left (4 A c e (2 c d-b e)+B \left (8 c^2 d^2+5 b^2 e^2-4 c e (3 b d+a e)\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{7/2}} \\ \end{align*}
Time = 2.24 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.24 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {-4 A c \left (3 b^3 e^3 x+b^2 e^2 (3 a e+c x (-6 d+e x))-2 b c \left (c d^2 (d-3 e x)+a e^2 (3 d+5 e x)\right )-4 c \left (2 a^2 e^3+c^2 d^3 x+a c e \left (-3 d^2-3 d e x+e^2 x^2\right )\right )\right )+B \left (4 a^2 c e^2 (-13 b e+6 c (4 d+e x))+b x \left (-8 c^3 d^3+15 b^3 e^3+b^2 c e^2 (-36 d+5 e x)-2 b c^2 e \left (-12 d^2+6 d e x+e^2 x^2\right )\right )+a \left (15 b^3 e^3-2 b^2 c e^2 (18 d+31 e x)+4 b c^2 e \left (6 d^2+30 d e x-5 e^2 x^2\right )-8 c^3 \left (2 d^3+6 d^2 e x-6 d e^2 x^2-e^3 x^3\right )\right )\right )}{4 c^3 \left (-b^2+4 a c\right ) \sqrt {a+x (b+c x)}}+\frac {3 e \left (4 A c e (-2 c d+b e)+B \left (-8 c^2 d^2-5 b^2 e^2+4 c e (3 b d+a e)\right )\right ) \log \left (c^3 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{8 c^{7/2}} \]
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Time = 0.74 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\frac {e^{2} \left (2 B c e x +4 A c e -7 B b e +12 B c d \right ) \sqrt {c \,x^{2}+b x +a}}{4 c^{3}}-\frac {-\frac {16 A \,c^{3} d^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 B \,e^{3} c \,a^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {14 B a \,b^{2} e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {8 A a b c \,e^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {24 B a b c d \,e^{2} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\left (12 A b \,c^{2} e^{3}-24 A \,c^{3} d \,e^{2}+12 B \,e^{3} a \,c^{2}-15 B \,e^{3} b^{2} c +36 B b \,c^{2} d \,e^{2}-24 B \,c^{3} d^{2} e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )+\left (8 A a \,c^{2} e^{3}+4 A \,b^{2} c \,e^{3}-24 A \,c^{3} d^{2} e -4 B a b c \,e^{3}+24 B a \,c^{2} d \,e^{2}-7 B \,b^{3} e^{3}+12 B \,b^{2} c d \,e^{2}-8 B \,c^{3} d^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{8 c^{3}}\) | \(554\) |
default | \(\frac {2 A \,d^{3} \left (2 c x +b \right )}{\left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+B \,e^{3} \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+b x +a}}-\frac {5 b \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )}{4 c}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )+\left (A \,e^{3}+3 B d \,e^{2}\right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+b x +a}}-\frac {3 b \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}-\frac {2 a \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{c}\right )+\left (3 A \,d^{2} e +B \,d^{3}\right ) \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{c^{\frac {3}{2}}}\right )\) | \(778\) |
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Leaf count of result is larger than twice the leaf count of optimal. 801 vs. \(2 (309) = 618\).
Time = 1.90 (sec) , antiderivative size = 1605, normalized size of antiderivative = 4.94 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{3}}{\left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 551, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (\frac {2 \, {\left (B b^{2} c^{2} e^{3} - 4 \, B a c^{3} e^{3}\right )} x}{b^{2} c^{3} - 4 \, a c^{4}} + \frac {12 \, B b^{2} c^{2} d e^{2} - 48 \, B a c^{3} d e^{2} - 5 \, B b^{3} c e^{3} + 20 \, B a b c^{2} e^{3} + 4 \, A b^{2} c^{2} e^{3} - 16 \, A a c^{3} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac {8 \, B b c^{3} d^{3} - 16 \, A c^{4} d^{3} - 24 \, B b^{2} c^{2} d^{2} e + 48 \, B a c^{3} d^{2} e + 24 \, A b c^{3} d^{2} e + 36 \, B b^{3} c d e^{2} - 120 \, B a b c^{2} d e^{2} - 24 \, A b^{2} c^{2} d e^{2} + 48 \, A a c^{3} d e^{2} - 15 \, B b^{4} e^{3} + 62 \, B a b^{2} c e^{3} + 12 \, A b^{3} c e^{3} - 24 \, B a^{2} c^{2} e^{3} - 40 \, A a b c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}\right )} x + \frac {16 \, B a c^{3} d^{3} - 8 \, A b c^{3} d^{3} - 24 \, B a b c^{2} d^{2} e + 48 \, A a c^{3} d^{2} e + 36 \, B a b^{2} c d e^{2} - 96 \, B a^{2} c^{2} d e^{2} - 24 \, A a b c^{2} d e^{2} - 15 \, B a b^{3} e^{3} + 52 \, B a^{2} b c e^{3} + 12 \, A a b^{2} c e^{3} - 32 \, A a^{2} c^{2} e^{3}}{b^{2} c^{3} - 4 \, a c^{4}}}{4 \, \sqrt {c x^{2} + b x + a}} - \frac {3 \, {\left (8 \, B c^{2} d^{2} e - 12 \, B b c d e^{2} + 8 \, A c^{2} d e^{2} + 5 \, B b^{2} e^{3} - 4 \, B a c e^{3} - 4 \, A b c e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8 \, c^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]
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