Integrand size = 29, antiderivative size = 229 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}-\frac {\left (3 a h^2 (3 f g+e h)-2 c g \left (f g^2+3 h (e g+d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \]
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Time = 0.20 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1659, 847, 794, 223, 212} \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=-\frac {h \sqrt {a+c x^2} \left (4 \left (4 a^2 f h^2-a c \left (3 h (d h+3 e g)+7 f g^2\right )+3 c^2 d g^2\right )+c h x (-9 a e h-11 a f g+6 c d g)\right )}{6 a c^3}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (-3 a h^2 (e h+3 f g)+6 c g h (d h+e g)+2 c f g^3\right )}{2 c^{5/2}}-\frac {h \sqrt {a+c x^2} (g+h x)^2 (3 c d-4 a f)}{3 a c^2}-\frac {(g+h x)^3 (a e-x (c d-a f))}{a c \sqrt {a+c x^2}} \]
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Rule 212
Rule 223
Rule 794
Rule 847
Rule 1659
Rubi steps \begin{align*} \text {integral}& = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {\int \frac {(g+h x)^2 (-a (f g+3 e h)+(3 c d-4 a f) h x)}{\sqrt {a+c x^2}} \, dx}{a c} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {\int \frac {(g+h x) \left (-a \left (2 (3 c d-4 a f) h^2+3 c g (f g+3 e h)\right )+c h (6 c d g-11 a f g-9 a e h) x\right )}{\sqrt {a+c x^2}} \, dx}{3 a c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{2 c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{2 c^2} \\ & = -\frac {(a e-(c d-a f) x) (g+h x)^3}{a c \sqrt {a+c x^2}}-\frac {(3 c d-4 a f) h (g+h x)^2 \sqrt {a+c x^2}}{3 a c^2}-\frac {h \left (4 \left (3 c^2 d g^2+4 a^2 f h^2-a c \left (7 f g^2+3 h (3 e g+d h)\right )\right )+c h (6 c d g-11 a f g-9 a e h) x\right ) \sqrt {a+c x^2}}{6 a c^3}+\frac {\left (2 c f g^3+6 c g h (e g+d h)-3 a h^2 (3 f g+e h)\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{2 c^{5/2}} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.12 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {-16 a^3 f h^3+6 c^3 d g^3 x+a c^2 \left (6 d h \left (-3 g^2-3 g h x+h^2 x^2\right )-3 e \left (2 g^3+6 g^2 h x-6 g h^2 x^2-h^3 x^3\right )+f x \left (-6 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )\right )+a^2 c h \left (f \left (36 g^2+27 g h x-8 h^2 x^2\right )+3 h (4 d h+3 e (4 g+h x))\right )+3 a \sqrt {c} \left (3 a h^2 (3 f g+e h)-2 c \left (f g^3+3 g h (e g+d h)\right )\right ) \sqrt {a+c x^2} \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{6 a c^3 \sqrt {a+c x^2}} \]
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Time = 0.85 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.24
method | result | size |
risch | \(-\frac {h \left (-2 f \,h^{2} c \,x^{2}-3 c e \,h^{2} x -9 c f g h x +10 a f \,h^{2}-6 c d \,h^{2}-18 c e g h -18 c f \,g^{2}\right ) \sqrt {c \,x^{2}+a}}{6 c^{3}}-\frac {\frac {a e \,h^{3} x}{\sqrt {c \,x^{2}+a}}-\frac {2 c^{2} d \,g^{3} x}{a \sqrt {c \,x^{2}+a}}+\frac {3 a f g \,h^{2} x}{\sqrt {c \,x^{2}+a}}+\left (3 a c e \,h^{3}+9 a c f g \,h^{2}-6 c^{2} d g \,h^{2}-6 c^{2} e \,g^{2} h -2 c^{2} f \,g^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )-\frac {-2 a^{2} f \,h^{3}+2 a c d \,h^{3}+6 a c e g \,h^{2}+6 a c f \,g^{2} h -6 c^{2} d \,g^{2} h -2 c^{2} e \,g^{3}}{c \sqrt {c \,x^{2}+a}}}{2 c^{2}}\) | \(285\) |
default | \(\frac {d \,g^{3} x}{a \sqrt {c \,x^{2}+a}}+f \,h^{3} \left (\frac {x^{4}}{3 c \sqrt {c \,x^{2}+a}}-\frac {4 a \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )}{3 c}\right )+\left (e \,h^{3}+3 f g \,h^{2}\right ) \left (\frac {x^{3}}{2 c \sqrt {c \,x^{2}+a}}-\frac {3 a \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )}{2 c}\right )-\frac {3 d \,g^{2} h +e \,g^{3}}{c \sqrt {c \,x^{2}+a}}+\left (d \,h^{3}+3 e g \,h^{2}+3 f \,g^{2} h \right ) \left (\frac {x^{2}}{c \sqrt {c \,x^{2}+a}}+\frac {2 a}{c^{2} \sqrt {c \,x^{2}+a}}\right )+\left (3 d g \,h^{2}+3 e \,g^{2} h +f \,g^{3}\right ) \left (-\frac {x}{c \sqrt {c \,x^{2}+a}}+\frac {\ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{c^{\frac {3}{2}}}\right )\) | \(292\) |
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Time = 0.32 (sec) , antiderivative size = 758, normalized size of antiderivative = 3.31 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\left [-\frac {3 \, {\left (2 \, a^{2} c f g^{3} + 6 \, a^{2} c e g^{2} h - 3 \, a^{3} e h^{3} + 3 \, {\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} g h^{2} + {\left (2 \, a c^{2} f g^{3} + 6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) - 2 \, {\left (2 \, a c^{2} f h^{3} x^{4} - 6 \, a c^{2} e g^{3} + 36 \, a^{2} c e g h^{2} - 18 \, {\left (a c^{2} d - 2 \, a^{2} c f\right )} g^{2} h + 4 \, {\left (3 \, a^{2} c d - 4 \, a^{3} f\right )} h^{3} + 3 \, {\left (3 \, a c^{2} f g h^{2} + a c^{2} e h^{3}\right )} x^{3} + 2 \, {\left (9 \, a c^{2} f g^{2} h + 9 \, a c^{2} e g h^{2} + {\left (3 \, a c^{2} d - 4 \, a^{2} c f\right )} h^{3}\right )} x^{2} - 3 \, {\left (6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} - 2 \, {\left (c^{3} d - a c^{2} f\right )} g^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{12 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}, -\frac {3 \, {\left (2 \, a^{2} c f g^{3} + 6 \, a^{2} c e g^{2} h - 3 \, a^{3} e h^{3} + 3 \, {\left (2 \, a^{2} c d - 3 \, a^{3} f\right )} g h^{2} + {\left (2 \, a c^{2} f g^{3} + 6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - {\left (2 \, a c^{2} f h^{3} x^{4} - 6 \, a c^{2} e g^{3} + 36 \, a^{2} c e g h^{2} - 18 \, {\left (a c^{2} d - 2 \, a^{2} c f\right )} g^{2} h + 4 \, {\left (3 \, a^{2} c d - 4 \, a^{3} f\right )} h^{3} + 3 \, {\left (3 \, a c^{2} f g h^{2} + a c^{2} e h^{3}\right )} x^{3} + 2 \, {\left (9 \, a c^{2} f g^{2} h + 9 \, a c^{2} e g h^{2} + {\left (3 \, a c^{2} d - 4 \, a^{2} c f\right )} h^{3}\right )} x^{2} - 3 \, {\left (6 \, a c^{2} e g^{2} h - 3 \, a^{2} c e h^{3} - 2 \, {\left (c^{3} d - a c^{2} f\right )} g^{3} + 3 \, {\left (2 \, a c^{2} d - 3 \, a^{2} c f\right )} g h^{2}\right )} x\right )} \sqrt {c x^{2} + a}}{6 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}}\right ] \]
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\[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {\left (g + h x\right )^{3} \left (d + e x + f x^{2}\right )}{\left (a + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.51 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {f h^{3} x^{4}}{3 \, \sqrt {c x^{2} + a} c} - \frac {4 \, a f h^{3} x^{2}}{3 \, \sqrt {c x^{2} + a} c^{2}} + \frac {d g^{3} x}{\sqrt {c x^{2} + a} a} - \frac {e g^{3}}{\sqrt {c x^{2} + a} c} - \frac {3 \, d g^{2} h}{\sqrt {c x^{2} + a} c} - \frac {8 \, a^{2} f h^{3}}{3 \, \sqrt {c x^{2} + a} c^{3}} + \frac {{\left (3 \, f g h^{2} + e h^{3}\right )} x^{3}}{2 \, \sqrt {c x^{2} + a} c} + \frac {{\left (3 \, f g^{2} h + 3 \, e g h^{2} + d h^{3}\right )} x^{2}}{\sqrt {c x^{2} + a} c} + \frac {3 \, {\left (3 \, f g h^{2} + e h^{3}\right )} a x}{2 \, \sqrt {c x^{2} + a} c^{2}} - \frac {{\left (f g^{3} + 3 \, e g^{2} h + 3 \, d g h^{2}\right )} x}{\sqrt {c x^{2} + a} c} - \frac {3 \, {\left (3 \, f g h^{2} + e h^{3}\right )} a \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, c^{\frac {5}{2}}} + \frac {{\left (f g^{3} + 3 \, e g^{2} h + 3 \, d g h^{2}\right )} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, f g^{2} h + 3 \, e g h^{2} + d h^{3}\right )} a}{\sqrt {c x^{2} + a} c^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45 \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {{\left ({\left ({\left (\frac {2 \, f h^{3} x}{c} + \frac {3 \, {\left (3 \, a c^{4} f g h^{2} + a c^{4} e h^{3}\right )}}{a c^{5}}\right )} x + \frac {2 \, {\left (9 \, a c^{4} f g^{2} h + 9 \, a c^{4} e g h^{2} + 3 \, a c^{4} d h^{3} - 4 \, a^{2} c^{3} f h^{3}\right )}}{a c^{5}}\right )} x + \frac {3 \, {\left (2 \, c^{5} d g^{3} - 2 \, a c^{4} f g^{3} - 6 \, a c^{4} e g^{2} h - 6 \, a c^{4} d g h^{2} + 9 \, a^{2} c^{3} f g h^{2} + 3 \, a^{2} c^{3} e h^{3}\right )}}{a c^{5}}\right )} x - \frac {2 \, {\left (3 \, a c^{4} e g^{3} + 9 \, a c^{4} d g^{2} h - 18 \, a^{2} c^{3} f g^{2} h - 18 \, a^{2} c^{3} e g h^{2} - 6 \, a^{2} c^{3} d h^{3} + 8 \, a^{3} c^{2} f h^{3}\right )}}{a c^{5}}}{6 \, \sqrt {c x^{2} + a}} - \frac {{\left (2 \, c f g^{3} + 6 \, c e g^{2} h + 6 \, c d g h^{2} - 9 \, a f g h^{2} - 3 \, a e h^{3}\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{2 \, c^{\frac {5}{2}}} \]
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Timed out. \[ \int \frac {(g+h x)^3 \left (d+e x+f x^2\right )}{\left (a+c x^2\right )^{3/2}} \, dx=\int \frac {{\left (g+h\,x\right )}^3\,\left (f\,x^2+e\,x+d\right )}{{\left (c\,x^2+a\right )}^{3/2}} \,d x \]
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