Integrand size = 26, antiderivative size = 392 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {\left (8 A c e \left (8 c^2 d^2-10 b c d e+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 d^2 e+8 b^2 c d e^2+3 b^3 e^3\right )-2 c e \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c^2 e^4}-\frac {(8 B c d-3 b B e-8 A c e-6 B c e x) \left (b x+c x^2\right )^{3/2}}{24 c e^2}-\frac {\left (8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )-B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{5/2} e^5}-\frac {d^{3/2} (B d-A e) (c d-b e)^{3/2} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^5} \]
-1/24*(-6*B*c*e*x-8*A*c*e-3*B*b*e+8*B*c*d)*(c*x^2+b*x)^(3/2)/c/e^2-1/64*(8 *A*c*e*(b^3*e^3+6*b^2*c*d*e^2-24*b*c^2*d^2*e+16*c^3*d^3)-B*(3*b^4*e^4+8*b^ 3*c*d*e^3+48*b^2*c^2*d^2*e^2-192*b*c^3*d^3*e+128*c^4*d^4))*arctanh(x*c^(1/ 2)/(c*x^2+b*x)^(1/2))/c^(5/2)/e^5-d^(3/2)*(-A*e+B*d)*(-b*e+c*d)^(3/2)*arct anh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/e ^5+1/64*(8*A*c*e*(b^2*e^2-10*b*c*d*e+8*c^2*d^2)-B*(3*b^3*e^3+8*b^2*c*d*e^2 -80*b*c^2*d^2*e+64*c^3*d^3)-2*c*e*(8*A*c*e*(-b*e+2*c*d)-B*(-3*b^2*e^2-8*b* c*d*e+16*c^2*d^2))*x)*(c*x^2+b*x)^(1/2)/c^2/e^4
Result contains complex when optimal does not.
Time = 5.19 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.71 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(x (b+c x))^{3/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (8 A c e \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+B \left (-9 b^3 e^3+6 b^2 c e^2 (-4 d+e x)+8 b c^2 e \left (30 d^2-14 d e x+9 e^2 x^2\right )-16 c^3 \left (12 d^3-6 d^2 e x+4 d e^2 x^2-3 e^3 x^3\right )\right )\right )+384 c^{3/2} \sqrt {d} (B d-A e) (c d-b e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+384 c^{3/2} \sqrt {d} (B d-A e) (c d-b e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+6 \left (-8 A c e \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right )+B \left (128 c^4 d^4-192 b c^3 d^3 e+48 b^2 c^2 d^2 e^2+8 b^3 c d e^3+3 b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{192 c^{5/2} e^5 x^{3/2} (b+c x)^{3/2}} \]
((x*(b + c*x))^(3/2)*(Sqrt[c]*e*Sqrt[x]*Sqrt[b + c*x]*(8*A*c*e*(3*b^2*e^2 + 2*b*c*e*(-15*d + 7*e*x) + 4*c^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + B*(-9*b ^3*e^3 + 6*b^2*c*e^2*(-4*d + e*x) + 8*b*c^2*e*(30*d^2 - 14*d*e*x + 9*e^2*x ^2) - 16*c^3*(12*d^3 - 6*d^2*e*x + 4*d*e^2*x^2 - 3*e^3*x^3))) + 384*c^(3/2 )*Sqrt[d]*(B*d - A*e)*(c*d - b*e)*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTa n[(Sqrt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/( Sqrt[d]*(-Sqrt[b] + Sqrt[b + c*x]))] + 384*c^(3/2)*Sqrt[d]*(B*d - A*e)*(c* d - b*e)*(c*d - b*e + I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b *e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[b] + Sqrt[ b + c*x]))] + 6*(-8*A*c*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b ^3*e^3) + B*(128*c^4*d^4 - 192*b*c^3*d^3*e + 48*b^2*c^2*d^2*e^2 + 8*b^3*c* d*e^3 + 3*b^4*e^4))*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]) )/(192*c^(5/2)*e^5*x^(3/2)*(b + c*x)^(3/2))
Time = 0.89 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.06, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1231, 27, 1231, 27, 1269, 1091, 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.
\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle -\frac {\int -\frac {\left (b d (8 B c d-3 b B e-8 A c e)-\left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)}dx}{8 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\left (b d (8 B c d-3 b B e-8 A c e)-\left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+8 b^2 c e^2 d+3 b^3 e^3\right )\right )+\left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c e d-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (8 c^2 d^2-10 b c e d+b^2 e^2\right )-B \left (64 c^3 d^3-80 b c^2 e d^2+8 b^2 c e^2 d+3 b^3 e^3\right )\right )+\left (4 b c d e (2 c d-b e) (8 B c d-3 b B e-8 A c e)+\left (8 c^2 d^2-4 b c e d-b^2 e^2\right ) \left (8 A c e (2 c d-b e)-B \left (16 c^2 d^2-8 b c e d-3 b^2 e^2\right )\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {\left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}+\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 1091 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}+\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {128 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}-\frac {256 c^2 d^2 (B d-A e) (c d-b e)^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+8 A c e \left (b^2 e^2-10 b c d e+8 c^2 d^2\right )-B \left (3 b^3 e^3+8 b^2 c d e^2-80 b c^2 d^2 e+64 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (\left (-b^2 e^2-4 b c d e+8 c^2 d^2\right ) \left (8 A c e (2 c d-b e)-B \left (-3 b^2 e^2-8 b c d e+16 c^2 d^2\right )\right )+4 b c d e (2 c d-b e) (-8 A c e-3 b B e+8 B c d)\right )}{\sqrt {c} e}+\frac {128 c^2 d^{3/2} (B d-A e) (c d-b e)^{3/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}}{16 c e^2}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c e-3 b B e+8 B c d-6 B c e x)}{24 c e^2}\) |
-1/24*((8*B*c*d - 3*b*B*e - 8*A*c*e - 6*B*c*e*x)*(b*x + c*x^2)^(3/2))/(c*e ^2) + (((8*A*c*e*(8*c^2*d^2 - 10*b*c*d*e + b^2*e^2) - B*(64*c^3*d^3 - 80*b *c^2*d^2*e + 8*b^2*c*d*e^2 + 3*b^3*e^3) - 2*c*e*(8*A*c*e*(2*c*d - b*e) - B *(16*c^2*d^2 - 8*b*c*d*e - 3*b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(4*c*e^2) - ( (2*(4*b*c*d*e*(2*c*d - b*e)*(8*B*c*d - 3*b*B*e - 8*A*c*e) + (8*c^2*d^2 - 4 *b*c*d*e - b^2*e^2)*(8*A*c*e*(2*c*d - b*e) - B*(16*c^2*d^2 - 8*b*c*d*e - 3 *b^2*e^2)))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) + (128*c^2 *d^(3/2)*(B*d - A*e)*(c*d - b*e)^(3/2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2* Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e)/(8*c*e^2))/(16*c*e^2)
3.12.75.3.1 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[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{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 = 0.83 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {-16 d^{2} \left (c^{\frac {5}{2}} b^{2} e^{2}-2 c^{\frac {7}{2}} b d e +c^{\frac {9}{2}} d^{2}\right ) \left (A e -B d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )+\sqrt {d \left (b e -c d \right )}\, \left (\left (\left (-A \,b^{3} c +\frac {3}{8} b^{4} B \right ) e^{4}+\left (-6 A \,b^{2} c^{2}+B \,b^{3} c \right ) d \,e^{3}+6 \left (4 A b \,c^{3}+B \,b^{2} c^{2}\right ) d^{2} e^{2}-16 c^{3} d^{3} \left (A c +\frac {3 B b}{2}\right ) e +16 B \,c^{4} d^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )+\left (2 \left (\left (B \,x^{3}+\frac {4}{3} A \,x^{2}\right ) e^{3}-2 x d \left (\frac {2 B x}{3}+A \right ) e^{2}+4 \left (\frac {B x}{2}+A \right ) d^{2} e -4 B \,d^{3}\right ) c^{\frac {7}{2}}+\left (\left (\left (3 B \,x^{2}+\frac {14}{3} A x \right ) e^{2}-10 d \left (\frac {7 B x}{15}+A \right ) e +10 B \,d^{2}\right ) c^{\frac {5}{2}}+\left (\left (\left (\frac {B x}{4}+A \right ) e -B d \right ) c^{\frac {3}{2}}-\frac {3 B b e \sqrt {c}}{8}\right ) e b \right ) e b \right ) \sqrt {x \left (c x +b \right )}\, e \right )}{8 \sqrt {d \left (b e -c d \right )}\, c^{\frac {5}{2}} e^{5}}\) | \(344\) |
risch | \(\frac {\left (48 B \,c^{3} e^{3} x^{3}+64 A \,c^{3} e^{3} x^{2}+72 B b \,c^{2} e^{3} x^{2}-64 B \,c^{3} d \,e^{2} x^{2}+112 A b \,c^{2} e^{3} x -96 A \,c^{3} d \,e^{2} x +6 B \,b^{2} c \,e^{3} x -112 B b \,c^{2} d \,e^{2} x +96 B \,c^{3} d^{2} e x +24 A \,b^{2} c \,e^{3}-240 A b \,c^{2} d \,e^{2}+192 A \,c^{3} d^{2} e -9 B \,b^{3} e^{3}-24 B \,b^{2} c d \,e^{2}+240 B b \,c^{2} d^{2} e -192 B \,c^{3} d^{3}\right ) x \left (c x +b \right )}{192 c^{2} e^{4} \sqrt {x \left (c x +b \right )}}-\frac {\frac {128 d^{2} \left (A \,b^{2} e^{3}-2 A b c d \,e^{2}+A \,c^{2} d^{2} e -B \,b^{2} d \,e^{2}+2 B b c \,d^{2} e -B \,c^{2} d^{3}\right ) c^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {\left (8 A \,b^{3} c \,e^{4}+48 A \,b^{2} c^{2} d \,e^{3}-192 A b \,c^{3} d^{2} e^{2}+128 A \,c^{4} d^{3} e -3 B \,b^{4} e^{4}-8 B \,b^{3} c d \,e^{3}-48 B \,b^{2} c^{2} d^{2} e^{2}+192 B b \,c^{3} d^{3} e -128 B \,c^{4} d^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}}{128 e^{4} c^{2}}\) | \(537\) |
default | \(\frac {B \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{e}+\frac {\left (A e -B d \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 {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \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 {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{2 e}-\frac {d \left (b e -c d \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 {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{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 {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{e^{2}}\right )}{e^{2}}\) | \(649\) |
1/8*(-16*d^2*(c^(5/2)*b^2*e^2-2*c^(7/2)*b*d*e+c^(9/2)*d^2)*(A*e-B*d)*arcta n((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+(d*(b*e-c*d))^(1/2)*(((-A*b^3 *c+3/8*b^4*B)*e^4+(-6*A*b^2*c^2+B*b^3*c)*d*e^3+6*(4*A*b*c^3+B*b^2*c^2)*d^2 *e^2-16*c^3*d^3*(A*c+3/2*B*b)*e+16*B*c^4*d^4)*arctanh((x*(c*x+b))^(1/2)/x/ c^(1/2))+(2*((B*x^3+4/3*A*x^2)*e^3-2*x*d*(2/3*B*x+A)*e^2+4*(1/2*B*x+A)*d^2 *e-4*B*d^3)*c^(7/2)+(((3*B*x^2+14/3*A*x)*e^2-10*d*(7/15*B*x+A)*e+10*B*d^2) *c^(5/2)+(((1/4*B*x+A)*e-B*d)*c^(3/2)-3/8*B*b*e*c^(1/2))*e*b)*e*b)*(x*(c*x +b))^(1/2)*e))/(d*(b*e-c*d))^(1/2)/c^(5/2)/e^5
Time = 23.46 (sec) , antiderivative size = 1695, normalized size of antiderivative = 4.32 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Too large to display} \]
[-1/384*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e^3 + (3*B*b^4 - 8*A*b ^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 384*(B*c ^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*sqrt(c*d^2 - b*d*e)*log( (b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d )) - 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c^4)*d^2* e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4*d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)*x)*sqrt( c*x^2 + b*x))/(c^3*e^5), -1/384*(768*(B*c^4*d^3 + A*b*c^3*d*e^2 - (B*b*c^3 + A*c^4)*d^2*e)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c* x^2 + b*x)/((c*d - b*e)*x)) + 3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)* d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*A*b^2*c^2)*d*e ^3 + (3*B*b^4 - 8*A*b^3*c)*e^4)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x )*sqrt(c)) - 2*(48*B*c^4*e^4*x^3 - 192*B*c^4*d^3*e + 48*(5*B*b*c^3 + 4*A*c ^4)*d^2*e^2 - 24*(B*b^2*c^2 + 10*A*b*c^3)*d*e^3 - 3*(3*B*b^3*c - 8*A*b^2*c ^2)*e^4 - 8*(8*B*c^4*d*e^3 - (9*B*b*c^3 + 8*A*c^4)*e^4)*x^2 + 2*(48*B*c^4* d^2*e^2 - 8*(7*B*b*c^3 + 6*A*c^4)*d*e^3 + (3*B*b^2*c^2 + 56*A*b*c^3)*e^4)* x)*sqrt(c*x^2 + b*x))/(c^3*e^5), -1/192*(3*(128*B*c^4*d^4 - 64*(3*B*b*c^3 + 2*A*c^4)*d^3*e + 48*(B*b^2*c^2 + 4*A*b*c^3)*d^2*e^2 + 8*(B*b^3*c - 6*...
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \]