Integrand size = 27, antiderivative size = 213 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (2 a f h^2+c \left (5 f g^2-7 h (e g+d h)\right )\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {a^2 (6 c d g-a f g-a e h) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1668, 794, 201, 223, 212} \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) (-a e h-a f g+6 c d g)}{16 c^{3/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (6 \left (2 a f h^2-7 c h (d h+e g)+5 c f g^2\right )+5 c h x (5 f g-7 e h)\right )}{210 c^2 h}+\frac {x \left (a+c x^2\right )^{3/2} (6 c d g-a (e h+f g))}{24 c}+\frac {a x \sqrt {a+c x^2} (-a e h-a f g+6 c d g)}{16 c}+\frac {f \left (a+c x^2\right )^{5/2} (g+h x)^2}{7 c h} \]
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Rule 201
Rule 212
Rule 223
Rule 794
Rule 1668
Rubi steps \begin{align*} \text {integral}& = \frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}+\frac {\int (g+h x) \left ((7 c d-2 a f) h^2-c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c h^2} \\ & = \frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {(6 c d g-a f g-a e h) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c} \\ & = \frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {(a (6 c d g-a f g-a e h)) \int \sqrt {a+c x^2} \, dx}{8 c} \\ & = \frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {\left (a^2 (6 c d g-a f g-a e h)\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{16 c} \\ & = \frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {\left (a^2 (6 c d g-a f g-a e h)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{16 c} \\ & = \frac {a (6 c d g-a f g-a e h) x \sqrt {a+c x^2}}{16 c}+\frac {(6 c d g-a (f g+e h)) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac {f (g+h x)^2 \left (a+c x^2\right )^{5/2}}{7 c h}-\frac {\left (6 \left (5 c f g^2+2 a f h^2-7 c h (e g+d h)\right )+5 c h (5 f g-7 e h) x\right ) \left (a+c x^2\right )^{5/2}}{210 c^2 h}+\frac {a^2 (6 c d g-a f g-a e h) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{16 c^{3/2}} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.92 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {\sqrt {a+c x^2} \left (-96 a^3 f h+3 a^2 c (112 d h+7 e (16 g+5 h x)+f x (35 g+16 h x))+4 c^3 x^3 (21 d (5 g+4 h x)+2 x (7 e (6 g+5 h x)+5 f x (7 g+6 h x)))+2 a c^2 x (21 d (25 g+16 h x)+x (7 e (48 g+35 h x)+f x (245 g+192 h x)))\right )+105 a^2 \sqrt {c} (-6 c d g+a f g+a e h) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{1680 c^2} \]
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Time = 0.53 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.91
method | result | size |
default | \(d g \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )+h f \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+\left (e h +f g \right ) \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {\left (d h +e g \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}\) | \(194\) |
risch | \(-\frac {\left (-240 c^{3} f h \,x^{6}-280 c^{3} e h \,x^{5}-280 c^{3} f g \,x^{5}-384 a \,c^{2} f h \,x^{4}-336 c^{3} d h \,x^{4}-336 c^{3} e g \,x^{4}-490 a \,c^{2} e h \,x^{3}-490 a \,c^{2} f g \,x^{3}-420 c^{3} d g \,x^{3}-48 a^{2} c f h \,x^{2}-672 a \,c^{2} d h \,x^{2}-672 a \,c^{2} e g \,x^{2}-105 a^{2} c e h x -105 a^{2} c f g x -1050 a \,c^{2} d g x +96 a^{3} f h -336 a^{2} c d h -336 a^{2} c e g \right ) \sqrt {c \,x^{2}+a}}{1680 c^{2}}-\frac {a^{2} \left (a e h +a f g -6 c d g \right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}\) | \(231\) |
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Time = 0.29 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.24 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\left [\frac {105 \, {\left (a^{3} e h - {\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (240 \, c^{3} f h x^{6} + 280 \, {\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \, {\left (7 \, c^{3} e g + {\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \, {\left (7 \, a c^{2} e h + {\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \, {\left (14 \, a c^{2} e g + {\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \, {\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \, {\left (a^{2} c e h + {\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt {c x^{2} + a}}{3360 \, c^{2}}, \frac {105 \, {\left (a^{3} e h - {\left (6 \, a^{2} c d - a^{3} f\right )} g\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (240 \, c^{3} f h x^{6} + 280 \, {\left (c^{3} f g + c^{3} e h\right )} x^{5} + 336 \, a^{2} c e g + 48 \, {\left (7 \, c^{3} e g + {\left (7 \, c^{3} d + 8 \, a c^{2} f\right )} h\right )} x^{4} + 70 \, {\left (7 \, a c^{2} e h + {\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} g\right )} x^{3} + 48 \, {\left (14 \, a c^{2} e g + {\left (14 \, a c^{2} d + a^{2} c f\right )} h\right )} x^{2} + 48 \, {\left (7 \, a^{2} c d - 2 \, a^{3} f\right )} h + 105 \, {\left (a^{2} c e h + {\left (10 \, a c^{2} d + a^{2} c f\right )} g\right )} x\right )} \sqrt {c x^{2} + a}}{1680 \, c^{2}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (196) = 392\).
Time = 0.59 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.38 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\begin {cases} \sqrt {a + c x^{2}} \left (\frac {c f h x^{6}}{7} + \frac {x^{5} \left (c^{2} e h + c^{2} f g\right )}{6 c} + \frac {x^{4} \cdot \left (\frac {8 a c f h}{7} + c^{2} d h + c^{2} e g\right )}{5 c} + \frac {x^{3} \cdot \left (2 a c e h + 2 a c f g - \frac {5 a \left (c^{2} e h + c^{2} f g\right )}{6 c} + c^{2} d g\right )}{4 c} + \frac {x^{2} \left (a^{2} f h + 2 a c d h + 2 a c e g - \frac {4 a \left (\frac {8 a c f h}{7} + c^{2} d h + c^{2} e g\right )}{5 c}\right )}{3 c} + \frac {x \left (a^{2} e h + a^{2} f g + 2 a c d g - \frac {3 a \left (2 a c e h + 2 a c f g - \frac {5 a \left (c^{2} e h + c^{2} f g\right )}{6 c} + c^{2} d g\right )}{4 c}\right )}{2 c} + \frac {a^{2} d h + a^{2} e g - \frac {2 a \left (a^{2} f h + 2 a c d h + 2 a c e g - \frac {4 a \left (\frac {8 a c f h}{7} + c^{2} d h + c^{2} e g\right )}{5 c}\right )}{3 c}}{c}\right ) + \left (a^{2} d g - \frac {a \left (a^{2} e h + a^{2} f g + 2 a c d g - \frac {3 a \left (2 a c e h + 2 a c f g - \frac {5 a \left (c^{2} e h + c^{2} f g\right )}{6 c} + c^{2} d g\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {c} \sqrt {a + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: a \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {c x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c \neq 0 \\a^{\frac {3}{2}} \left (d g x + \frac {f h x^{4}}{4} + \frac {x^{3} \left (e h + f g\right )}{3} + \frac {x^{2} \left (d h + e g\right )}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.99 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f h x^{2}}{7 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d g x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d g x + \frac {3 \, a^{2} d g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} e g}{5 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} d h}{5 \, c} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} a f h}{35 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} {\left (f g + e h\right )} x}{6 \, c} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (f g + e h\right )} a x}{24 \, c} - \frac {\sqrt {c x^{2} + a} {\left (f g + e h\right )} a^{2} x}{16 \, c} - \frac {{\left (f g + e h\right )} a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} \]
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Time = 0.30 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.21 \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\frac {1}{1680} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, c f h x + \frac {7 \, {\left (c^{6} f g + c^{6} e h\right )}}{c^{5}}\right )} x + \frac {6 \, {\left (7 \, c^{6} e g + 7 \, c^{6} d h + 8 \, a c^{5} f h\right )}}{c^{5}}\right )} x + \frac {35 \, {\left (6 \, c^{6} d g + 7 \, a c^{5} f g + 7 \, a c^{5} e h\right )}}{c^{5}}\right )} x + \frac {24 \, {\left (14 \, a c^{5} e g + 14 \, a c^{5} d h + a^{2} c^{4} f h\right )}}{c^{5}}\right )} x + \frac {105 \, {\left (10 \, a c^{5} d g + a^{2} c^{4} f g + a^{2} c^{4} e h\right )}}{c^{5}}\right )} x + \frac {48 \, {\left (7 \, a^{2} c^{4} e g + 7 \, a^{2} c^{4} d h - 2 \, a^{3} c^{3} f h\right )}}{c^{5}}\right )} - \frac {{\left (6 \, a^{2} c d g - a^{3} f g - a^{3} e h\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int (g+h x) \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx=\int \left (g+h\,x\right )\,{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \]
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