Optimal. Leaf size=346 \[ \frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (8 a h^2 (2 f g+e h)+c g \left (5 f g^2-8 h (e g+7 d h)\right )\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.32, antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {1668, 847, 794,
201, 223, 212} \begin {gather*} \frac {x \left (a+c x^2\right )^{3/2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{192 c^2}+\frac {a x \sqrt {a+c x^2} \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^2}+\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f h^2-8 a c \left (h (d h+2 e g)+f g^2\right )+48 c^2 d g^2\right )}{128 c^{5/2}}-\frac {\left (a+c x^2\right )^{5/2} \left (12 \left (8 a h^2 (e h+2 f g)-8 c g h (7 d h+e g)+5 c f g^3\right )-5 h x \left (7 h^2 (8 c d-3 a f)-2 c g (5 f g-8 e h)\right )\right )}{1680 c^2 h}-\frac {\left (a+c x^2\right )^{5/2} (g+h x)^2 (5 f g-8 e h)}{56 c h}+\frac {f \left (a+c x^2\right )^{5/2} (g+h x)^3}{8 c h} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 212
Rule 223
Rule 794
Rule 847
Rule 1668
Rubi steps
\begin {align*} \int (g+h x)^2 \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac {\int (g+h x)^2 \left ((8 c d-3 a f) h^2-c h (5 f g-8 e h) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{8 c h^2}\\ &=-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}+\frac {\int (g+h x) \left (c h^2 (56 c d g-11 a f g-16 a e h)+c h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{3/2} \, dx}{56 c^2 h^2}\\ &=-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{48 c^2}\\ &=\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \sqrt {a+c x^2} \, dx}{64 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \int \frac {1}{\sqrt {a+c x^2}} \, dx}{128 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {\left (a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {a+c x^2}}\right )}{128 c^2}\\ &=\frac {a \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \sqrt {a+c x^2}}{128 c^2}+\frac {\left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) x \left (a+c x^2\right )^{3/2}}{192 c^2}-\frac {(5 f g-8 e h) (g+h x)^2 \left (a+c x^2\right )^{5/2}}{56 c h}+\frac {f (g+h x)^3 \left (a+c x^2\right )^{5/2}}{8 c h}-\frac {\left (12 \left (5 c f g^3-8 c g h (e g+7 d h)+8 a h^2 (2 f g+e h)\right )-5 h \left (7 (8 c d-3 a f) h^2-2 c g (5 f g-8 e h)\right ) x\right ) \left (a+c x^2\right )^{5/2}}{1680 c^2 h}+\frac {a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{128 c^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.09, size = 332, normalized size = 0.96 \begin {gather*} \frac {\sqrt {c} \sqrt {a+c x^2} \left (-3 a^3 h (512 f g+256 e h+105 f h x)+6 a^2 c \left (28 d h (32 g+5 h x)+8 e \left (56 g^2+35 g h x+8 h^2 x^2\right )+f x \left (140 g^2+128 g h x+35 h^2 x^2\right )\right )+16 c^3 x^3 \left (14 d \left (15 g^2+24 g h x+10 h^2 x^2\right )+x \left (8 e \left (21 g^2+35 g h x+15 h^2 x^2\right )+5 f x \left (28 g^2+48 g h x+21 h^2 x^2\right )\right )\right )+8 a c^2 x \left (14 d \left (75 g^2+96 g h x+35 h^2 x^2\right )+x \left (4 e \left (168 g^2+245 g h x+96 h^2 x^2\right )+f x \left (490 g^2+768 g h x+315 h^2 x^2\right )\right )\right )\right )-105 a^2 \left (48 c^2 d g^2+3 a^2 f h^2-8 a c \left (f g^2+h (2 e g+d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{13440 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 320, normalized size = 0.92
method | result | size |
default | \(f \,h^{2} \left (\frac {x^{3} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{8 c}-\frac {3 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )}{8 c}\right )+\left (e \,h^{2}+2 f g h \right ) \left (\frac {x^{2} \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{7 c}-\frac {2 a \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{35 c^{2}}\right )+\left (d \,h^{2}+2 e g h +f \,g^{2}\right ) \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6 c}-\frac {a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6 c}\right )+\frac {\left (2 d g h +g^{2} e \right ) \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5 c}+d \,g^{2} \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )\) | \(320\) |
risch | \(-\frac {\left (-1680 c^{3} f \,h^{2} x^{7}-1920 c^{3} e \,h^{2} x^{6}-3840 c^{3} f g h \,x^{6}-2520 a \,c^{2} f \,h^{2} x^{5}-2240 c^{3} d \,h^{2} x^{5}-4480 c^{3} e g h \,x^{5}-2240 c^{3} f \,g^{2} x^{5}-3072 a \,c^{2} e \,h^{2} x^{4}-6144 a \,c^{2} f g h \,x^{4}-5376 c^{3} d g h \,x^{4}-2688 c^{3} e \,g^{2} x^{4}-210 a^{2} f \,h^{2} c \,x^{3}-3920 a \,c^{2} d \,h^{2} x^{3}-7840 a \,c^{2} e g h \,x^{3}-3920 a \,c^{2} f \,g^{2} x^{3}-3360 c^{3} d \,g^{2} x^{3}-384 a^{2} c e \,h^{2} x^{2}-768 a^{2} c f g h \,x^{2}-10752 a \,c^{2} d g h \,x^{2}-5376 a \,c^{2} e \,g^{2} x^{2}+315 a^{3} f \,h^{2} x -840 a^{2} c d \,h^{2} x -1680 a^{2} c e g h x -840 a^{2} c f \,g^{2} x -8400 a \,c^{2} d \,g^{2} x +768 a^{3} e \,h^{2}+1536 a^{3} f g h -5376 a^{2} c d g h -2688 a^{2} c e \,g^{2}\right ) \sqrt {c \,x^{2}+a}}{13440 c^{2}}+\frac {3 a^{4} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) f \,h^{2}}{128 c^{\frac {5}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d \,h^{2}}{16 c^{\frac {3}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) e g h}{8 c^{\frac {3}{2}}}-\frac {a^{3} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) f \,g^{2}}{16 c^{\frac {3}{2}}}+\frac {3 a^{2} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right ) d \,g^{2}}{8 \sqrt {c}}\) | \(489\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 387, normalized size = 1.12 \begin {gather*} \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} f h^{2} x^{3}}{8 \, c} + \frac {1}{4} \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} d g^{2} x + \frac {3}{8} \, \sqrt {c x^{2} + a} a d g^{2} x - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} a f h^{2} x}{16 \, c^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} a^{2} f h^{2} x}{64 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a^{3} f h^{2} x}{128 \, c^{2}} + \frac {3 \, a^{2} d g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c}} + \frac {3 \, a^{4} f h^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{128 \, c^{\frac {5}{2}}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} d g h}{5 \, c} + \frac {{\left (2 \, f g h + h^{2} e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} x^{2}}{7 \, c} + \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} g^{2} e}{5 \, c} + \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} x}{6 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} {\left (c x^{2} + a\right )}^{\frac {3}{2}} a x}{24 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} \sqrt {c x^{2} + a} a^{2} x}{16 \, c} - \frac {{\left (f g^{2} + d h^{2} + 2 \, g h e\right )} a^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{16 \, c^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, f g h + h^{2} e\right )} {\left (c x^{2} + a\right )}^{\frac {5}{2}} a}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, size = 851, normalized size = 2.46 \begin {gather*} \left [\frac {105 \, {\left (16 \, a^{3} c g h e - 8 \, {\left (6 \, a^{2} c^{2} d - a^{3} c f\right )} g^{2} + {\left (8 \, a^{3} c d - 3 \, a^{4} f\right )} h^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 2 \, {\left (1680 \, c^{4} f h^{2} x^{7} + 3840 \, c^{4} f g h x^{6} + 768 \, {\left (7 \, c^{4} d + 8 \, a c^{3} f\right )} g h x^{4} + 280 \, {\left (8 \, c^{4} f g^{2} + {\left (8 \, c^{4} d + 9 \, a c^{3} f\right )} h^{2}\right )} x^{5} + 768 \, {\left (14 \, a c^{3} d + a^{2} c^{2} f\right )} g h x^{2} + 70 \, {\left (8 \, {\left (6 \, c^{4} d + 7 \, a c^{3} f\right )} g^{2} + {\left (56 \, a c^{3} d + 3 \, a^{2} c^{2} f\right )} h^{2}\right )} x^{3} + 768 \, {\left (7 \, a^{2} c^{2} d - 2 \, a^{3} c f\right )} g h + 105 \, {\left (8 \, {\left (10 \, a c^{3} d + a^{2} c^{2} f\right )} g^{2} + {\left (8 \, a^{2} c^{2} d - 3 \, a^{3} c f\right )} h^{2}\right )} x + 16 \, {\left (120 \, c^{4} h^{2} x^{6} + 280 \, c^{4} g h x^{5} + 490 \, a c^{3} g h x^{3} + 105 \, a^{2} c^{2} g h x + 168 \, a^{2} c^{2} g^{2} - 48 \, a^{3} c h^{2} + 24 \, {\left (7 \, c^{4} g^{2} + 8 \, a c^{3} h^{2}\right )} x^{4} + 24 \, {\left (14 \, a c^{3} g^{2} + a^{2} c^{2} h^{2}\right )} x^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{26880 \, c^{3}}, \frac {105 \, {\left (16 \, a^{3} c g h e - 8 \, {\left (6 \, a^{2} c^{2} d - a^{3} c f\right )} g^{2} + {\left (8 \, a^{3} c d - 3 \, a^{4} f\right )} h^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) + {\left (1680 \, c^{4} f h^{2} x^{7} + 3840 \, c^{4} f g h x^{6} + 768 \, {\left (7 \, c^{4} d + 8 \, a c^{3} f\right )} g h x^{4} + 280 \, {\left (8 \, c^{4} f g^{2} + {\left (8 \, c^{4} d + 9 \, a c^{3} f\right )} h^{2}\right )} x^{5} + 768 \, {\left (14 \, a c^{3} d + a^{2} c^{2} f\right )} g h x^{2} + 70 \, {\left (8 \, {\left (6 \, c^{4} d + 7 \, a c^{3} f\right )} g^{2} + {\left (56 \, a c^{3} d + 3 \, a^{2} c^{2} f\right )} h^{2}\right )} x^{3} + 768 \, {\left (7 \, a^{2} c^{2} d - 2 \, a^{3} c f\right )} g h + 105 \, {\left (8 \, {\left (10 \, a c^{3} d + a^{2} c^{2} f\right )} g^{2} + {\left (8 \, a^{2} c^{2} d - 3 \, a^{3} c f\right )} h^{2}\right )} x + 16 \, {\left (120 \, c^{4} h^{2} x^{6} + 280 \, c^{4} g h x^{5} + 490 \, a c^{3} g h x^{3} + 105 \, a^{2} c^{2} g h x + 168 \, a^{2} c^{2} g^{2} - 48 \, a^{3} c h^{2} + 24 \, {\left (7 \, c^{4} g^{2} + 8 \, a c^{3} h^{2}\right )} x^{4} + 24 \, {\left (14 \, a c^{3} g^{2} + a^{2} c^{2} h^{2}\right )} x^{2}\right )} e\right )} \sqrt {c x^{2} + a}}{13440 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 49.78, size = 1304, normalized size = 3.77 \begin {gather*} - \frac {3 a^{\frac {7}{2}} f h^{2} x}{128 c^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} d h^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} e g h x}{8 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {5}{2}} f g^{2} x}{16 c \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {a^{\frac {5}{2}} f h^{2} x^{3}}{128 c \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {a^{\frac {3}{2}} d g^{2} x \sqrt {1 + \frac {c x^{2}}{a}}}{2} + \frac {a^{\frac {3}{2}} d g^{2} x}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} d h^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} e g h x^{3}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {17 a^{\frac {3}{2}} f g^{2} x^{3}}{48 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {13 a^{\frac {3}{2}} f h^{2} x^{5}}{64 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 \sqrt {a} c d g^{2} x^{3}}{8 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c d h^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c e g h x^{5}}{12 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {11 \sqrt {a} c f g^{2} x^{5}}{24 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {5 \sqrt {a} c f h^{2} x^{7}}{16 \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {3 a^{4} f h^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{128 c^{\frac {5}{2}}} - \frac {a^{3} d h^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} - \frac {a^{3} e g h \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 c^{\frac {3}{2}}} - \frac {a^{3} f g^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{16 c^{\frac {3}{2}}} + \frac {3 a^{2} d g^{2} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )}}{8 \sqrt {c}} + 2 a d g h \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a e g^{2} \left (\begin {cases} \frac {\sqrt {a} x^{2}}{2} & \text {for}\: c = 0 \\\frac {\left (a + c x^{2}\right )^{\frac {3}{2}}}{3 c} & \text {otherwise} \end {cases}\right ) + a e h^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 a f g h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + 2 c d g h \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c e g^{2} \left (\begin {cases} - \frac {2 a^{2} \sqrt {a + c x^{2}}}{15 c^{2}} + \frac {a x^{2} \sqrt {a + c x^{2}}}{15 c} + \frac {x^{4} \sqrt {a + c x^{2}}}{5} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases}\right ) + c e h^{2} \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + 2 c f g h \left (\begin {cases} \frac {8 a^{3} \sqrt {a + c x^{2}}}{105 c^{3}} - \frac {4 a^{2} x^{2} \sqrt {a + c x^{2}}}{105 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{2}}}{35 c} + \frac {x^{6} \sqrt {a + c x^{2}}}{7} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{6}}{6} & \text {otherwise} \end {cases}\right ) + \frac {c^{2} d g^{2} x^{5}}{4 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} d h^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} e g h x^{7}}{3 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} f g^{2} x^{7}}{6 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {c^{2} f h^{2} x^{9}}{8 \sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 5.04, size = 452, normalized size = 1.31 \begin {gather*} \frac {1}{13440} \, \sqrt {c x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, c f h^{2} x + \frac {8 \, {\left (2 \, c^{7} f g h + c^{7} h^{2} e\right )}}{c^{6}}\right )} x + \frac {7 \, {\left (8 \, c^{7} f g^{2} + 8 \, c^{7} d h^{2} + 9 \, a c^{6} f h^{2} + 16 \, c^{7} g h e\right )}}{c^{6}}\right )} x + \frac {48 \, {\left (14 \, c^{7} d g h + 16 \, a c^{6} f g h + 7 \, c^{7} g^{2} e + 8 \, a c^{6} h^{2} e\right )}}{c^{6}}\right )} x + \frac {35 \, {\left (48 \, c^{7} d g^{2} + 56 \, a c^{6} f g^{2} + 56 \, a c^{6} d h^{2} + 3 \, a^{2} c^{5} f h^{2} + 112 \, a c^{6} g h e\right )}}{c^{6}}\right )} x + \frac {192 \, {\left (28 \, a c^{6} d g h + 2 \, a^{2} c^{5} f g h + 14 \, a c^{6} g^{2} e + a^{2} c^{5} h^{2} e\right )}}{c^{6}}\right )} x + \frac {105 \, {\left (80 \, a c^{6} d g^{2} + 8 \, a^{2} c^{5} f g^{2} + 8 \, a^{2} c^{5} d h^{2} - 3 \, a^{3} c^{4} f h^{2} + 16 \, a^{2} c^{5} g h e\right )}}{c^{6}}\right )} x + \frac {384 \, {\left (14 \, a^{2} c^{5} d g h - 4 \, a^{3} c^{4} f g h + 7 \, a^{2} c^{5} g^{2} e - 2 \, a^{3} c^{4} h^{2} e\right )}}{c^{6}}\right )} - \frac {{\left (48 \, a^{2} c^{2} d g^{2} - 8 \, a^{3} c f g^{2} - 8 \, a^{3} c d h^{2} + 3 \, a^{4} f h^{2} - 16 \, a^{3} c g h e\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right )}{128 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (g+h\,x\right )}^2\,{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________