Integrand size = 27, antiderivative size = 441 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=-\frac {\left (3 b^3 e^3 g-64 c^3 d^2 (e f-d g)+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (2 b e f-2 b d g+3 a e g)+2 c e \left (3 b^2 e^2 g+16 c^2 d (e f-d g)-4 c e (2 b e f-2 b d g+3 a e g)\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^2 e^4}+\frac {(8 c e f-8 c d g+3 b e g+6 c e g x) \left (a+b x+c x^2\right )^{3/2}}{24 c e^2}+\frac {\left (3 b^4 e^4 g-128 c^4 d^3 (e f-d g)+192 c^3 d e (b d-a e) (e f-d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g-b^2 d (e f-d g)+2 a b e (e f-d g)\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{5/2} e^5}+\frac {\left (c d^2-b d e+a e^2\right )^{3/2} (e f-d g) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \]
1/24*(6*c*e*g*x+3*b*e*g-8*c*d*g+8*c*e*f)*(c*x^2+b*x+a)^(3/2)/c/e^2+1/128*( 3*b^4*e^4*g-128*c^4*d^3*(-d*g+e*f)+192*c^3*d*e*(-a*e+b*d)*(-d*g+e*f)-8*b^2 *c*e^3*(3*a*e*g-b*d*g+b*e*f)+48*c^2*e^2*(a^2*e^2*g-b^2*d*(-d*g+e*f)+2*a*b* e*(-d*g+e*f)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)/ e^5+(a*e^2-b*d*e+c*d^2)^(3/2)*(-d*g+e*f)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c* d)*x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e^5-1/64*(3*b^3*e^3*g -64*c^3*d^2*(-d*g+e*f)+16*c^2*e*(-4*a*e+5*b*d)*(-d*g+e*f)-4*b*c*e^2*(3*a*e *g-2*b*d*g+2*b*e*f)+2*c*e*(3*b^2*e^2*g+16*c^2*d*(-d*g+e*f)-4*c*e*(3*a*e*g- 2*b*d*g+2*b*e*f))*x)*(c*x^2+b*x+a)^(1/2)/c^2/e^4
Time = 2.48 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.97 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (-9 b^3 e^3 g-16 c^3 \left (12 d^3 g-6 d^2 e (2 f+g x)+2 d e^2 x (3 f+2 g x)-e^3 x^2 (4 f+3 g x)\right )+6 b c e^2 (10 a e g+b (4 e f-4 d g+e g x))+8 c^2 e \left (a e (32 e f-32 d g+15 e g x)+b \left (30 d^2 g-2 d e (15 f+7 g x)+e^2 x (14 f+9 g x)\right )\right )\right )}{c^2}-768 \sqrt {-c d^2+b d e-a e^2} \left (c d^2+e (-b d+a e)\right ) (-e f+d g) \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-\frac {3 \left (3 b^4 e^4 g+128 c^4 d^3 (-e f+d g)-192 c^3 d e (b d-a e) (-e f+d g)-8 b^2 c e^3 (b e f-b d g+3 a e g)+48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 d (-e f+d g)\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{5/2}}}{384 e^5} \]
((2*e*Sqrt[a + x*(b + c*x)]*(-9*b^3*e^3*g - 16*c^3*(12*d^3*g - 6*d^2*e*(2* f + g*x) + 2*d*e^2*x*(3*f + 2*g*x) - e^3*x^2*(4*f + 3*g*x)) + 6*b*c*e^2*(1 0*a*e*g + b*(4*e*f - 4*d*g + e*g*x)) + 8*c^2*e*(a*e*(32*e*f - 32*d*g + 15* e*g*x) + b*(30*d^2*g - 2*d*e*(15*f + 7*g*x) + e^2*x*(14*f + 9*g*x)))))/c^2 - 768*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(c*d^2 + e*(-(b*d) + a*e))*(-(e*f) + d*g)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] - (3*(3*b^4*e^4*g + 128*c^4*d^3*(-(e*f) + d*g) - 192*c^3* d*e*(b*d - a*e)*(-(e*f) + d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e*g) + 4 8*c^2*e^2*(a^2*e^2*g + 2*a*b*e*(e*f - d*g) + b^2*d*(-(e*f) + d*g)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2))/(384*e^5)
Time = 1.14 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1231, 27, 1231, 27, 1269, 1092, 219, 1154, 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.
| \(\displaystyle \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\int \frac {\left (3 d e g b^2+8 c d (e f-d g) b-4 a c e (4 e f-d g)+\left (16 d (e f-d g) c^2-4 e (2 b e f-2 b d g+3 a e g) c+3 b^2 e^2 g\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{8 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\int \frac {\left (3 d e g b^2+8 c d (e f-d g) b-4 a c e (4 e f-d g)+\left (16 d (e f-d g) c^2-4 e (2 b e f-2 b d g+3 a e g) c+3 b^2 e^2 g\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{16 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (3 d e g b^2+8 c d (e f-d g) b-4 a c e (4 e f-d g)\right )-d \left (-e b^2+4 c d b-4 a c e\right ) \left (16 d (e f-d g) c^2-4 e (2 b e f-2 b d g+3 a e g) c+3 b^2 e^2 g\right )+\left (-128 d^3 (e f-d g) c^4+192 d e (b d-a e) (e f-d g) c^3+48 e^2 \left (-d (e f-d g) b^2+2 a e (e f-d g) b+a^2 e^2 g\right ) c^2-8 b^2 e^3 (b e f-b d g+3 a e g) c+3 b^4 e^4 g\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (3 d e g b^2+8 c d (e f-d g) b-4 a c e (4 e f-d g)\right )-d \left (-e b^2+4 c d b-4 a c e\right ) \left (16 d (e f-d g) c^2-4 e (2 b e f-2 b d g+3 a e g) c+3 b^2 e^2 g\right )+\left (-128 d^3 (e f-d g) c^4+192 d e (b d-a e) (e f-d g) c^3+48 e^2 \left (-d (e f-d g) b^2+2 a e (e f-d g) b+a^2 e^2 g\right ) c^2-8 b^2 e^3 (b e f-b d g+3 a e g) c+3 b^4 e^4 g\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\frac {\left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}+\frac {128 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\frac {2 \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}+\frac {128 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\frac {128 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right )}{\sqrt {c} e}-\frac {256 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^2 \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}}{8 c e^2}}{16 c e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a+b x+c x^2\right )^{3/2} (3 b e g-8 c d g+8 c e f+6 c e g x)}{24 c e^2}-\frac {\frac {\sqrt {a+b x+c x^2} \left (2 c e x \left (-4 c e (3 a e g-2 b d g+2 b e f)+3 b^2 e^2 g+16 c^2 d (e f-d g)\right )+16 c^2 e (5 b d-4 a e) (e f-d g)-4 b c e^2 (3 a e g-2 b d g+2 b e f)+3 b^3 e^3 g-64 c^3 d^2 (e f-d g)\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (48 c^2 e^2 \left (a^2 e^2 g+2 a b e (e f-d g)+b^2 (-d) (e f-d g)\right )-8 b^2 c e^3 (3 a e g-b d g+b e f)+192 c^3 d e (b d-a e) (e f-d g)+3 b^4 e^4 g-128 c^4 d^3 (e f-d g)\right )}{\sqrt {c} e}+\frac {128 c^2 (e f-d g) \left (a e^2-b d e+c d^2\right )^{3/2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}}{16 c e^2}\) |
((8*c*e*f - 8*c*d*g + 3*b*e*g + 6*c*e*g*x)*(a + b*x + c*x^2)^(3/2))/(24*c* e^2) - (((3*b^3*e^3*g - 64*c^3*d^2*(e*f - d*g) + 16*c^2*e*(5*b*d - 4*a*e)* (e*f - d*g) - 4*b*c*e^2*(2*b*e*f - 2*b*d*g + 3*a*e*g) + 2*c*e*(3*b^2*e^2*g + 16*c^2*d*(e*f - d*g) - 4*c*e*(2*b*e*f - 2*b*d*g + 3*a*e*g))*x)*Sqrt[a + b*x + c*x^2])/(4*c*e^2) - (((3*b^4*e^4*g - 128*c^4*d^3*(e*f - d*g) + 192* c^3*d*e*(b*d - a*e)*(e*f - d*g) - 8*b^2*c*e^3*(b*e*f - b*d*g + 3*a*e*g) + 48*c^2*e^2*(a^2*e^2*g - b^2*d*(e*f - d*g) + 2*a*b*e*(e*f - d*g)))*ArcTanh[ (b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) + (128*c^2*(c* d^2 - b*d*e + a*e^2)^(3/2)*(e*f - d*g)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e )*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e)/(8*c*e^2)) /(16*c*e^2)
3.9.64.3.1 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_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.73 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(\frac {\left (48 g \,e^{3} x^{3} c^{3}+72 b \,c^{2} e^{3} g \,x^{2}-64 c^{3} d \,e^{2} g \,x^{2}+64 c^{3} e^{3} f \,x^{2}+120 a \,c^{2} e^{3} g x +6 b^{2} c \,e^{3} g x -112 b \,c^{2} d \,e^{2} g x +112 b \,c^{2} e^{3} f x +96 c^{3} d^{2} e g x -96 c^{3} d \,e^{2} f x +60 a b c \,e^{3} g -256 a \,c^{2} d \,e^{2} g +256 a \,c^{2} e^{3} f -9 b^{3} e^{3} g -24 b^{2} c d \,e^{2} g +24 b^{2} c \,e^{3} f +240 b \,c^{2} d^{2} e g -240 b \,c^{2} d \,e^{2} f -192 c^{3} d^{3} g +192 d^{2} f \,c^{3} e \right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{2} e^{4}}+\frac {\frac {128 \left (a^{2} d \,e^{4} g -f \,a^{2} e^{5}-2 a b \,d^{2} e^{3} g +2 a b d \,e^{4} f +2 a c \,d^{3} e^{2} g -2 a c \,d^{2} e^{3} f +b^{2} d^{3} e^{2} g -b^{2} d^{2} e^{3} f -2 b c \,d^{4} e g +2 b c \,d^{3} e^{2} f +d^{5} c^{2} g -c^{2} d^{4} e f \right ) c^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (48 a^{2} c^{2} e^{4} g -24 a \,b^{2} c \,e^{4} g -96 a b \,c^{2} d \,e^{3} g +96 a b \,c^{2} e^{4} f +192 a \,c^{3} d^{2} e^{2} g -192 a \,c^{3} d \,e^{3} f +3 b^{4} e^{4} g +8 b^{3} c d \,e^{3} g -8 b^{3} c \,e^{4} f +48 b^{2} c^{2} d^{2} e^{2} g -48 b^{2} c^{2} d \,e^{3} f -192 b \,c^{3} d^{3} e g +192 b \,c^{3} d^{2} e^{2} f +128 c^{4} d^{4} g -128 d^{3} f \,c^{4} e \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{128 e^{4} c^{2}}\) | \(735\) |
| default | \(\frac {g \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{e}+\frac {\left (-d g +e f \right ) \left (\frac {\left (\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (b e -2 c d \right ) \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{4 c}+\frac {\left (\frac {4 c \left (e^{2} a -b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}+\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\right )}{e^{2}}\) | \(742\) |
1/192/c^2*(48*c^3*e^3*g*x^3+72*b*c^2*e^3*g*x^2-64*c^3*d*e^2*g*x^2+64*c^3*e ^3*f*x^2+120*a*c^2*e^3*g*x+6*b^2*c*e^3*g*x-112*b*c^2*d*e^2*g*x+112*b*c^2*e ^3*f*x+96*c^3*d^2*e*g*x-96*c^3*d*e^2*f*x+60*a*b*c*e^3*g-256*a*c^2*d*e^2*g+ 256*a*c^2*e^3*f-9*b^3*e^3*g-24*b^2*c*d*e^2*g+24*b^2*c*e^3*f+240*b*c^2*d^2* e*g-240*b*c^2*d*e^2*f-192*c^3*d^3*g+192*c^3*d^2*e*f)*(c*x^2+b*x+a)^(1/2)/e ^4+1/128/e^4/c^2*(128*(a^2*d*e^4*g-a^2*e^5*f-2*a*b*d^2*e^3*g+2*a*b*d*e^4*f +2*a*c*d^3*e^2*g-2*a*c*d^2*e^3*f+b^2*d^3*e^2*g-b^2*d^2*e^3*f-2*b*c*d^4*e*g +2*b*c*d^3*e^2*f+c^2*d^5*g-c^2*d^4*e*f)*c^2/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^ (1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+ c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e ^2)^(1/2))/(x+d/e))+(48*a^2*c^2*e^4*g-24*a*b^2*c*e^4*g-96*a*b*c^2*d*e^3*g+ 96*a*b*c^2*e^4*f+192*a*c^3*d^2*e^2*g-192*a*c^3*d*e^3*f+3*b^4*e^4*g+8*b^3*c *d*e^3*g-8*b^3*c*e^4*f+48*b^2*c^2*d^2*e^2*g-48*b^2*c^2*d*e^3*f-192*b*c^3*d ^3*e*g+192*b*c^3*d^2*e^2*f+128*c^4*d^4*g-128*c^4*d^3*e*f)/e*ln((1/2*b+c*x) /c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2))
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \]
\[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (f + g x\right ) \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}}{d+e\,x} \,d x \]