Integrand size = 29, antiderivative size = 432 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=-\frac {\left (8 \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right )-h \left (20 c f g^2-16 c e g h+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt {a+c x^2}}{8 h^5}-\frac {\left (4 \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right )-3 h \left (a f h^2+c \left (5 f g^2-4 h (e g-d h)\right )\right ) x\right ) \left (a+c x^2\right )^{3/2}}{12 h^3 \left (c g^2+a h^2\right )}-\frac {\left (f g^2-e g h+d h^2\right ) \left (a+c x^2\right )^{5/2}}{h \left (c g^2+a h^2\right ) (g+h x)}+\frac {\left (3 a^2 f h^4+8 c^2 g^2 \left (5 f g^2-h (4 e g-3 d h)\right )+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )}{8 \sqrt {c} h^6}+\frac {\sqrt {c g^2+a h^2} \left (a h^2 (2 f g-e h)+c g \left (5 f g^2-h (4 e g-3 d h)\right )\right ) \text {arctanh}\left (\frac {a h-c g x}{\sqrt {c g^2+a h^2} \sqrt {a+c x^2}}\right )}{h^6} \]
-1/12*(4*a*h^2*(-e*h+2*f*g)+4*c*g*(5*f*g^2-h*(-3*d*h+4*e*g))-3*h*(a*f*h^2+ c*(5*f*g^2-4*h*(-d*h+e*g)))*x)*(c*x^2+a)^(3/2)/h^3/(a*h^2+c*g^2)-(d*h^2-e* g*h+f*g^2)*(c*x^2+a)^(5/2)/h/(a*h^2+c*g^2)/(h*x+g)+1/8*(3*a^2*f*h^4+8*c^2* g^2*(5*f*g^2-h*(-3*d*h+4*e*g))+12*a*c*h^2*(3*f*g^2-h*(-d*h+2*e*g)))*arctan h(x*c^(1/2)/(c*x^2+a)^(1/2))/h^6/c^(1/2)+(a*h^2*(-e*h+2*f*g)+c*g*(5*f*g^2- h*(-3*d*h+4*e*g)))*arctanh((-c*g*x+a*h)/(a*h^2+c*g^2)^(1/2)/(c*x^2+a)^(1/2 ))*(a*h^2+c*g^2)^(1/2)/h^6-1/8*(8*a*h^2*(-e*h+2*f*g)+8*c*g*(5*f*g^2-h*(-3* d*h+4*e*g))-h*(3*a*f*h^2+12*c*d*h^2-16*c*e*g*h+20*c*f*g^2)*x)*(c*x^2+a)^(1 /2)/h^5
Time = 1.49 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\frac {\frac {h \sqrt {a+c x^2} \left (-2 c f \left (60 g^4+30 g^3 h x-10 g^2 h^2 x^2+5 g h^3 x^3-3 h^4 x^4\right )+a h^2 \left (8 h (7 e g-3 d h+4 e h x)+f \left (-88 g^2-49 g h x+15 h^2 x^2\right )\right )+4 c h \left (3 d h \left (-6 g^2-3 g h x+h^2 x^2\right )+2 e \left (12 g^3+6 g^2 h x-2 g h^2 x^2+h^3 x^3\right )\right )\right )}{g+h x}-48 \sqrt {-c g^2-a h^2} \left (5 c f g^3+c g h (-4 e g+3 d h)+a h^2 (2 f g-e h)\right ) \arctan \left (\frac {\sqrt {c} (g+h x)-h \sqrt {a+c x^2}}{\sqrt {-c g^2-a h^2}}\right )-\frac {3 \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2+h (-2 e g+d h)\right )+8 c^2 \left (5 f g^4+g^2 h (-4 e g+3 d h)\right )\right ) \log \left (-\sqrt {c} x+\sqrt {a+c x^2}\right )}{\sqrt {c}}}{24 h^6} \]
((h*Sqrt[a + c*x^2]*(-2*c*f*(60*g^4 + 30*g^3*h*x - 10*g^2*h^2*x^2 + 5*g*h^ 3*x^3 - 3*h^4*x^4) + a*h^2*(8*h*(7*e*g - 3*d*h + 4*e*h*x) + f*(-88*g^2 - 4 9*g*h*x + 15*h^2*x^2)) + 4*c*h*(3*d*h*(-6*g^2 - 3*g*h*x + h^2*x^2) + 2*e*( 12*g^3 + 6*g^2*h*x - 2*g*h^2*x^2 + h^3*x^3))))/(g + h*x) - 48*Sqrt[-(c*g^2 ) - a*h^2]*(5*c*f*g^3 + c*g*h*(-4*e*g + 3*d*h) + a*h^2*(2*f*g - e*h))*ArcT an[(Sqrt[c]*(g + h*x) - h*Sqrt[a + c*x^2])/Sqrt[-(c*g^2) - a*h^2]] - (3*(3 *a^2*f*h^4 + 12*a*c*h^2*(3*f*g^2 + h*(-2*e*g + d*h)) + 8*c^2*(5*f*g^4 + g^ 2*h*(-4*e*g + 3*d*h)))*Log[-(Sqrt[c]*x) + Sqrt[a + c*x^2]])/Sqrt[c])/(24*h ^6)
Time = 0.98 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {2182, 25, 682, 25, 27, 682, 27, 719, 224, 219, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx\) |
| \(\Big \downarrow \) 2182 |
| \(\displaystyle -\frac {\int -\frac {\left (c d g-a f g+a e h+\left (a f h-c \left (-\frac {5 f g^2}{h}+4 e g-4 d h\right )\right ) x\right ) \left (c x^2+a\right )^{3/2}}{g+h x}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (c d g-a f g+a e h+\left (a f h-c \left (-\frac {5 f g^2}{h}+4 e g-4 d h\right )\right ) x\right ) \left (c x^2+a\right )^{3/2}}{g+h x}dx}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {\frac {\int -\frac {c \left (c g^2+a h^2\right ) \left (a h (5 f g-4 e h)-\left (20 c f g^2-16 c e h g+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt {c x^2+a}}{h (g+h x)}dx}{4 c h^2}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {c \left (c g^2+a h^2\right ) \left (a h (5 f g-4 e h)-\left (20 c f g^2-16 c e h g+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt {c x^2+a}}{h (g+h x)}dx}{4 c h^2}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \int \frac {\left (a h (5 f g-4 e h)-\left (20 c f g^2-16 c e h g+12 c d h^2+3 a f h^2\right ) x\right ) \sqrt {c x^2+a}}{g+h x}dx}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {\int \frac {c \left (a h \left (a (13 f g-8 e h) h^2+4 c \left (5 f g^3-g h (4 e g-3 d h)\right )\right )-\left (2 a c g (5 f g-4 e h) h^2+\left (2 c g^2+a h^2\right ) \left (3 a f h^2+4 c \left (5 f g^2-h (4 e g-3 d h)\right )\right )\right ) x\right )}{(g+h x) \sqrt {c x^2+a}}dx}{2 c h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {\int \frac {a h \left (a (13 f g-8 e h) h^2+4 c \left (5 f g^3-g h (4 e g-3 d h)\right )\right )-\left (2 a c g (5 f g-4 e h) h^2+\left (2 c g^2+a h^2\right ) \left (3 a f h^2+4 c \left (5 f g^2-h (4 e g-3 d h)\right )\right )\right ) x}{(g+h x) \sqrt {c x^2+a}}dx}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {\frac {8 \left (a h^2+c g^2\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right ) \int \frac {1}{\sqrt {c x^2+a}}dx}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {\frac {8 \left (a h^2+c g^2\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {\frac {8 \left (a h^2+c g^2\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right ) \int \frac {1}{(g+h x) \sqrt {c x^2+a}}dx}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right )}{\sqrt {c} h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {-\frac {8 \left (a h^2+c g^2\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right ) \int \frac {1}{c g^2+a h^2-\frac {(a h-c g x)^2}{c x^2+a}}d\frac {a h-c g x}{\sqrt {c x^2+a}}}{h}-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right )}{\sqrt {c} h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {\left (a h^2+c g^2\right ) \left (\frac {-\frac {\text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right ) \left (3 a^2 f h^4+12 a c h^2 \left (3 f g^2-h (2 e g-d h)\right )+8 c^2 \left (5 f g^4-g^2 h (4 e g-3 d h)\right )\right )}{\sqrt {c} h}-\frac {8 \sqrt {a h^2+c g^2} \text {arctanh}\left (\frac {a h-c g x}{\sqrt {a+c x^2} \sqrt {a h^2+c g^2}}\right ) \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )}{h}}{2 h^2}+\frac {\sqrt {a+c x^2} \left (8 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-h x \left (3 a f h^2+12 c d h^2-16 c e g h+20 c f g^2\right )\right )}{2 h^2}\right )}{4 h^3}-\frac {\left (a+c x^2\right )^{3/2} \left (4 \left (a h^2 (2 f g-e h)-c g h (4 e g-3 d h)+5 c f g^3\right )-3 h x \left (a f h^2-4 c h (e g-d h)+5 c f g^2\right )\right )}{12 h^3}}{a h^2+c g^2}-\frac {\left (a+c x^2\right )^{5/2} \left (d h^2-e g h+f g^2\right )}{h (g+h x) \left (a h^2+c g^2\right )}\) |
-(((f*g^2 - e*g*h + d*h^2)*(a + c*x^2)^(5/2))/(h*(c*g^2 + a*h^2)*(g + h*x) )) + (-1/12*((4*(5*c*f*g^3 - c*g*h*(4*e*g - 3*d*h) + a*h^2*(2*f*g - e*h)) - 3*h*(5*c*f*g^2 + a*f*h^2 - 4*c*h*(e*g - d*h))*x)*(a + c*x^2)^(3/2))/h^3 - ((c*g^2 + a*h^2)*(((8*(5*c*f*g^3 - c*g*h*(4*e*g - 3*d*h) + a*h^2*(2*f*g - e*h)) - h*(20*c*f*g^2 - 16*c*e*g*h + 12*c*d*h^2 + 3*a*f*h^2)*x)*Sqrt[a + c*x^2])/(2*h^2) + (-(((3*a^2*f*h^4 + 8*c^2*(5*f*g^4 - g^2*h*(4*e*g - 3*d* h)) + 12*a*c*h^2*(3*f*g^2 - h*(2*e*g - d*h)))*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(Sqrt[c]*h)) - (8*Sqrt[c*g^2 + a*h^2]*(5*c*f*g^3 - c*g*h*(4*e*g - 3*d*h) + a*h^2*(2*f*g - e*h))*ArcTanh[(a*h - c*g*x)/(Sqrt[c*g^2 + a*h^2] *Sqrt[a + c*x^2])])/h)/(2*h^2)))/(4*h^3))/(c*g^2 + a*h^2)
3.1.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b *e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
Time = 0.74 (sec) , antiderivative size = 736, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(\frac {\left (6 f \,x^{3} h^{3} c +8 c e \,h^{3} x^{2}-16 c f g \,h^{2} x^{2}+15 a f \,h^{3} x +12 c d \,h^{3} x -24 c e g \,h^{2} x +36 c f \,g^{2} h x +32 a e \,h^{3}-64 a f g \,h^{2}-48 c d g \,h^{2}+72 c e \,g^{2} h -96 c f \,g^{3}\right ) \sqrt {c \,x^{2}+a}}{24 h^{5}}+\frac {\frac {\left (3 a^{2} f \,h^{4}+12 a c d \,h^{4}-24 a c e g \,h^{3}+36 a c f \,g^{2} h^{2}+24 c^{2} d \,g^{2} h^{2}-32 c^{2} e \,g^{3} h +40 c^{2} f \,g^{4}\right ) \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{h \sqrt {c}}-\frac {\left (8 a^{2} e \,h^{5}-16 a^{2} f g \,h^{4}-32 a c d g \,h^{4}+48 a c e \,g^{2} h^{3}-64 a c f \,g^{3} h^{2}-32 c^{2} d \,g^{3} h^{2}+40 c^{2} e \,g^{4} h -48 c^{2} f \,g^{5}\right ) \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{h^{2} \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}+\frac {\left (8 a^{2} d \,h^{6}-8 a^{2} e g \,h^{5}+8 a^{2} f \,g^{2} h^{4}+16 a c d \,g^{2} h^{4}-16 g^{3} a c e \,h^{3}+16 a c f \,g^{4} h^{2}+8 c^{2} d \,g^{4} h^{2}-8 g^{5} c^{2} e h +8 g^{6} c^{2} f \right ) \left (-\frac {h^{2} \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{\left (a \,h^{2}+c \,g^{2}\right ) \left (x +\frac {g}{h}\right )}-\frac {c g h \ln \left (\frac {\frac {2 a \,h^{2}+2 c \,g^{2}}{h^{2}}-\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+2 \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}\, \sqrt {\left (x +\frac {g}{h}\right )^{2} c -\frac {2 c g \left (x +\frac {g}{h}\right )}{h}+\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}{x +\frac {g}{h}}\right )}{\left (a \,h^{2}+c \,g^{2}\right ) \sqrt {\frac {a \,h^{2}+c \,g^{2}}{h^{2}}}}\right )}{h^{3}}}{8 h^{5}}\) | \(736\) |
| default | \(\text {Expression too large to display}\) | \(1448\) |
1/24*(6*c*f*h^3*x^3+8*c*e*h^3*x^2-16*c*f*g*h^2*x^2+15*a*f*h^3*x+12*c*d*h^3 *x-24*c*e*g*h^2*x+36*c*f*g^2*h*x+32*a*e*h^3-64*a*f*g*h^2-48*c*d*g*h^2+72*c *e*g^2*h-96*c*f*g^3)*(c*x^2+a)^(1/2)/h^5+1/8/h^5*((3*a^2*f*h^4+12*a*c*d*h^ 4-24*a*c*e*g*h^3+36*a*c*f*g^2*h^2+24*c^2*d*g^2*h^2-32*c^2*e*g^3*h+40*c^2*f *g^4)/h*ln(x*c^(1/2)+(c*x^2+a)^(1/2))/c^(1/2)-(8*a^2*e*h^5-16*a^2*f*g*h^4- 32*a*c*d*g*h^4+48*a*c*e*g^2*h^3-64*a*c*f*g^3*h^2-32*c^2*d*g^3*h^2+40*c^2*e *g^4*h-48*c^2*f*g^5)/h^2/((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2 -2*c*g/h*(x+1/h*g)+2*((a*h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1 /h*g)+(a*h^2+c*g^2)/h^2)^(1/2))/(x+1/h*g))+1/h^3*(8*a^2*d*h^6-8*a^2*e*g*h^ 5+8*a^2*f*g^2*h^4+16*a*c*d*g^2*h^4-16*a*c*e*g^3*h^3+16*a*c*f*g^4*h^2+8*c^2 *d*g^4*h^2-8*c^2*e*g^5*h+8*c^2*f*g^6)*(-1/(a*h^2+c*g^2)*h^2/(x+1/h*g)*((x+ 1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^(1/2)-c*g*h/(a*h^2+c*g^2)/ ((a*h^2+c*g^2)/h^2)^(1/2)*ln((2*(a*h^2+c*g^2)/h^2-2*c*g/h*(x+1/h*g)+2*((a* h^2+c*g^2)/h^2)^(1/2)*((x+1/h*g)^2*c-2*c*g/h*(x+1/h*g)+(a*h^2+c*g^2)/h^2)^ (1/2))/(x+1/h*g))))
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {\left (a + c x^{2}\right )^{\frac {3}{2}} \left (d + e x + f x^{2}\right )}{\left (g + h x\right )^{2}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=-\frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f g^{2}}{h^{4} x + g h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e g}{h^{3} x + g h^{2}} - \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} d}{h^{2} x + g h} + \frac {5 \, \sqrt {c x^{2} + a} c f g^{2} x}{2 \, h^{4}} - \frac {2 \, \sqrt {c x^{2} + a} c e g x}{h^{3}} + \frac {3 \, \sqrt {c x^{2} + a} c d x}{2 \, h^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} f x}{4 \, h^{2}} + \frac {3 \, \sqrt {c x^{2} + a} a f x}{8 \, h^{2}} + \frac {5 \, c^{\frac {3}{2}} f g^{4} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{6}} - \frac {4 \, c^{\frac {3}{2}} e g^{3} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{5}} + \frac {3 \, c^{\frac {3}{2}} d g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{4}} + \frac {9 \, a \sqrt {c} f g^{2} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{4}} - \frac {3 \, a \sqrt {c} e g \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{h^{3}} + \frac {3 \, a \sqrt {c} d \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{2 \, h^{2}} + \frac {3 \, a^{2} f \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {c} h^{2}} - \frac {3 \, \sqrt {a + \frac {c g^{2}}{h^{2}}} c f g^{3} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{5}} + \frac {3 \, \sqrt {a + \frac {c g^{2}}{h^{2}}} c e g^{2} \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{4}} - \frac {3 \, \sqrt {a + \frac {c g^{2}}{h^{2}}} c d g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} - \frac {2 \, {\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} f g \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{3}} + \frac {{\left (a + \frac {c g^{2}}{h^{2}}\right )}^{\frac {3}{2}} e \operatorname {arsinh}\left (\frac {c g x}{\sqrt {a c} {\left | h x + g \right |}} - \frac {a h}{\sqrt {a c} {\left | h x + g \right |}}\right )}{h^{2}} - \frac {5 \, \sqrt {c x^{2} + a} c f g^{3}}{h^{5}} + \frac {4 \, \sqrt {c x^{2} + a} c e g^{2}}{h^{4}} - \frac {3 \, \sqrt {c x^{2} + a} c d g}{h^{3}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} f g}{3 \, h^{3}} - \frac {2 \, \sqrt {c x^{2} + a} a f g}{h^{3}} + \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} e}{3 \, h^{2}} + \frac {\sqrt {c x^{2} + a} a e}{h^{2}} \]
-(c*x^2 + a)^(3/2)*f*g^2/(h^4*x + g*h^3) + (c*x^2 + a)^(3/2)*e*g/(h^3*x + g*h^2) - (c*x^2 + a)^(3/2)*d/(h^2*x + g*h) + 5/2*sqrt(c*x^2 + a)*c*f*g^2*x /h^4 - 2*sqrt(c*x^2 + a)*c*e*g*x/h^3 + 3/2*sqrt(c*x^2 + a)*c*d*x/h^2 + 1/4 *(c*x^2 + a)^(3/2)*f*x/h^2 + 3/8*sqrt(c*x^2 + a)*a*f*x/h^2 + 5*c^(3/2)*f*g ^4*arcsinh(c*x/sqrt(a*c))/h^6 - 4*c^(3/2)*e*g^3*arcsinh(c*x/sqrt(a*c))/h^5 + 3*c^(3/2)*d*g^2*arcsinh(c*x/sqrt(a*c))/h^4 + 9/2*a*sqrt(c)*f*g^2*arcsin h(c*x/sqrt(a*c))/h^4 - 3*a*sqrt(c)*e*g*arcsinh(c*x/sqrt(a*c))/h^3 + 3/2*a* sqrt(c)*d*arcsinh(c*x/sqrt(a*c))/h^2 + 3/8*a^2*f*arcsinh(c*x/sqrt(a*c))/(s qrt(c)*h^2) - 3*sqrt(a + c*g^2/h^2)*c*f*g^3*arcsinh(c*g*x/(sqrt(a*c)*abs(h *x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^5 + 3*sqrt(a + c*g^2/h^2)*c*e*g ^2*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/ h^4 - 3*sqrt(a + c*g^2/h^2)*c*d*g*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 - 2*(a + c*g^2/h^2)^(3/2)*f*g*arcsinh(c *g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)*abs(h*x + g)))/h^3 + (a + c *g^2/h^2)^(3/2)*e*arcsinh(c*g*x/(sqrt(a*c)*abs(h*x + g)) - a*h/(sqrt(a*c)* abs(h*x + g)))/h^2 - 5*sqrt(c*x^2 + a)*c*f*g^3/h^5 + 4*sqrt(c*x^2 + a)*c*e *g^2/h^4 - 3*sqrt(c*x^2 + a)*c*d*g/h^3 - 2/3*(c*x^2 + a)^(3/2)*f*g/h^3 - 2 *sqrt(c*x^2 + a)*a*f*g/h^3 + 1/3*(c*x^2 + a)^(3/2)*e/h^2 + sqrt(c*x^2 + a) *a*e/h^2
\[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e x + d\right )}}{{\left (h x + g\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right )}{(g+h x)^2} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{3/2}\,\left (f\,x^2+e\,x+d\right )}{{\left (g+h\,x\right )}^2} \,d x \]