Integrand size = 26, antiderivative size = 402 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=-\frac {b^2 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \sqrt {b x+c x^2}}{512 d^4 (c d-b e)^4 (d+e x)^2}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) (b d+(2 c d-b e) x) \left (b x+c x^2\right )^{3/2}}{192 d^3 (c d-b e)^3 (d+e x)^4}+\frac {(B d-A e) \left (b x+c x^2\right )^{5/2}}{6 d (c d-b e) (d+e x)^6}-\frac {(7 A e (2 c d-b e)-B d (2 c d+5 b e)) \left (b x+c x^2\right )^{5/2}}{60 d^2 (c d-b e)^2 (d+e x)^5}+\frac {b^4 \left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{1024 d^{9/2} (c d-b e)^{9/2}} \]
1/192*(24*A*c^2*d^2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*e+5*B*d))*(b*d+(-b*e+2 *c*d)*x)*(c*x^2+b*x)^(3/2)/d^3/(-b*e+c*d)^3/(e*x+d)^4+1/6*(-A*e+B*d)*(c*x^ 2+b*x)^(5/2)/d/(-b*e+c*d)/(e*x+d)^6-1/60*(7*A*e*(-b*e+2*c*d)-B*d*(5*b*e+2* c*d))*(c*x^2+b*x)^(5/2)/d^2/(-b*e+c*d)^2/(e*x+d)^5+1/1024*b^4*(24*A*c^2*d^ 2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*e+5*B*d))*arctanh(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))/d^(9/2)/(-b*e+c*d)^(9/2)-1/ 512*b^2*(24*A*c^2*d^2-12*b*c*d*(2*A*e+B*d)+b^2*e*(7*A*e+5*B*d))*(b*d+(-b*e +2*c*d)*x)*(c*x^2+b*x)^(1/2)/d^4/(-b*e+c*d)^4/(e*x+d)^2
Time = 10.94 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.88 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\frac {(x (b+c x))^{3/2} \left (\frac {(-B d+A e) x^{5/2} (b+c x)}{(d+e x)^6}-\frac {(7 A e (-2 c d+b e)+B d (2 c d+5 b e)) x^{5/2} (b+c x)}{10 d (c d-b e) (d+e x)^5}+\frac {\left (24 A c^2 d^2-12 b c d (B d+2 A e)+b^2 e (5 B d+7 A e)\right ) \left (-\frac {2 x^{3/2} (b+c x)^{5/2}}{(d+e x)^4}+\frac {b \sqrt {x} (b+c x)^{5/2}}{(c d-b e) (d+e x)^3}-\frac {b^2 \sqrt {x} \sqrt {b+c x} (5 b d+2 c d x+3 b e x)}{8 d^2 (c d-b e) (d+e x)^2}-\frac {3 b^4 \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )}{8 d^{5/2} (c d-b e)^{3/2}}\right )}{32 d (c d-b e)^2 (b+c x)^{3/2}}\right )}{6 d (-c d+b e) x^{3/2}} \]
((x*(b + c*x))^(3/2)*(((-(B*d) + A*e)*x^(5/2)*(b + c*x))/(d + e*x)^6 - ((7 *A*e*(-2*c*d + b*e) + B*d*(2*c*d + 5*b*e))*x^(5/2)*(b + c*x))/(10*d*(c*d - b*e)*(d + e*x)^5) + ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B* d + 7*A*e))*((-2*x^(3/2)*(b + c*x)^(5/2))/(d + e*x)^4 + (b*Sqrt[x]*(b + c* x)^(5/2))/((c*d - b*e)*(d + e*x)^3) - (b^2*Sqrt[x]*Sqrt[b + c*x]*(5*b*d + 2*c*d*x + 3*b*e*x))/(8*d^2*(c*d - b*e)*(d + e*x)^2) - (3*b^4*ArcTanh[(Sqrt [c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(8*d^(5/2)*(c*d - b*e)^(3/2 ))))/(32*d*(c*d - b*e)^2*(b + c*x)^(3/2))))/(6*d*(-(c*d) + b*e)*x^(3/2))
Time = 0.60 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1237, 27, 25, 1228, 1152, 1152, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\int -\frac {(12 A c d-b (5 B d+7 A e)+2 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{2 (d+e x)^6}dx}{6 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {(5 b B d-12 A c d+7 A b e-2 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^6}dx}{12 d (c d-b e)}+\frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\int \frac {(5 b B d-12 A c d+7 A b e-2 c (B d-A e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^6}dx}{12 d (c d-b e)}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{5 d (d+e x)^5 (c d-b e)}-\frac {\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \int \frac {\left (c x^2+b x\right )^{3/2}}{(d+e x)^5}dx}{2 d (c d-b e)}}{12 d (c d-b e)}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{5 d (d+e x)^5 (c d-b e)}-\frac {\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \int \frac {\sqrt {c x^2+b x}}{(d+e x)^3}dx}{16 d (c d-b e)}\right )}{2 d (c d-b e)}}{12 d (c d-b e)}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{5 d (d+e x)^5 (c d-b e)}-\frac {\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{8 d (c d-b e)}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}}{12 d (c d-b e)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{5 d (d+e x)^5 (c d-b e)}-\frac {\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {b^2 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{4 d (c d-b e)}+\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}}{12 d (c d-b e)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (B d-A e)}{6 d (d+e x)^6 (c d-b e)}-\frac {\frac {\left (b x+c x^2\right )^{5/2} (7 A e (2 c d-b e)-B d (5 b e+2 c d))}{5 d (d+e x)^5 (c d-b e)}-\frac {\left (b^2 e (7 A e+5 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right ) \left (\frac {\left (b x+c x^2\right )^{3/2} (x (2 c d-b e)+b d)}{8 d (d+e x)^4 (c d-b e)}-\frac {3 b^2 \left (\frac {\sqrt {b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac {b^2 \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}}\right )}{16 d (c d-b e)}\right )}{2 d (c d-b e)}}{12 d (c d-b e)}\) |
((B*d - A*e)*(b*x + c*x^2)^(5/2))/(6*d*(c*d - b*e)*(d + e*x)^6) - (((7*A*e *(2*c*d - b*e) - B*d*(2*c*d + 5*b*e))*(b*x + c*x^2)^(5/2))/(5*d*(c*d - b*e )*(d + e*x)^5) - ((24*A*c^2*d^2 - 12*b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 7*A*e))*(((b*d + (2*c*d - b*e)*x)*(b*x + c*x^2)^(3/2))/(8*d*(c*d - b*e)*(d + e*x)^4) - (3*b^2*(((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqr t[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8*d^(3/2)*(c*d - b*e)^(3/2))))/(16*d*(c *d - b*e))))/(2*d*(c*d - b*e)))/(12*d*(c*d - b*e))
3.12.81.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 1.10 (sec) , antiderivative size = 598, normalized size of antiderivative = 1.49
| method | result | size |
| pseudoelliptic | \(-\frac {7 \left (\left (\frac {12 \left (2 A \,c^{2}-B b c \right ) d^{2}}{7}-\frac {24 \left (A c -\frac {5 B b}{24}\right ) e b d}{7}+A \,b^{2} e^{2}\right ) \left (e x +d \right )^{6} b^{4} \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 (\frac {24 c \left (-\frac {b^{4} B}{2}+c \left (\frac {B x}{3}+A \right ) b^{3}-\frac {2 c^{2} x \left (\frac {2 B x}{5}+A \right ) b^{2}}{3}-8 c^{3} x^{2} \left (\frac {11 B x}{15}+A \right ) b -\frac {16 c^{4} \left (\frac {4 B x}{5}+A \right ) x^{3}}{3}\right ) d^{7}}{7}-\frac {24 \left (-\frac {5 B \,b^{5}}{24}+c \left (\frac {107 B x}{36}+A \right ) b^{4}-\frac {19 c^{2} x \left (\frac {89 B x}{285}+A \right ) b^{3}}{3}-\frac {448 c^{3} \left (\frac {171 B x}{224}+A \right ) x^{2} b^{2}}{15}-\frac {72 c^{4} x^{3} \left (\frac {80 B x}{81}+A \right ) b}{5}+\frac {32 c^{5} x^{4} \left (\frac {B x}{3}+A \right )}{15}\right ) e \,d^{6}}{7}+\left (\left (\frac {85 B x}{21}+A \right ) b^{5}-\frac {422 c x \left (\frac {1328 B x}{1055}+A \right ) b^{4}}{21}-\frac {2456 c^{2} \left (\frac {285 B x}{307}+A \right ) x^{2} b^{3}}{21}-\frac {816 c^{3} x^{3} \left (\frac {332 B x}{153}+A \right ) b^{2}}{35}+\frac {1984 c^{4} \left (\frac {13 B x}{31}+A \right ) x^{4} b}{105}-\frac {128 A \,c^{5} x^{5}}{105}\right ) e^{2} d^{5}+\frac {17 \left (\left (\frac {198 B x}{119}+A \right ) b^{4}+\frac {7144 c \left (\frac {69 B x}{94}+A \right ) x \,b^{3}}{595}-\frac {2232 c^{2} x^{2} \left (-\frac {37 B x}{93}+A \right ) b^{2}}{595}-\frac {736 c^{3} \left (\frac {51 B x}{46}+A \right ) x^{3} b}{595}+\frac {64 A \,c^{4} x^{4}}{119}\right ) x \,e^{3} b \,d^{4}}{3}-\frac {562 x^{2} \left (\left (\frac {165 B x}{281}+A \right ) b^{3}-\frac {642 c \left (\frac {185 B x}{963}+A \right ) x \,b^{2}}{281}+\frac {168 c^{2} x^{2} \left (-\frac {5 B x}{63}+A \right ) b}{281}+\frac {16 A \,c^{3} x^{3}}{281}\right ) e^{4} b^{2} d^{3}}{35}-\frac {66 x^{3} \left (\left (\frac {425 B x}{1386}+A \right ) b^{2}-\frac {824 c \left (\frac {65 B x}{824}+A \right ) x b}{693}+\frac {8 A \,c^{2} x^{2}}{63}\right ) e^{5} b^{3} d^{2}}{5}-\frac {17 x^{4} \left (\left (\frac {15 B x}{119}+A \right ) b -\frac {58 A c x}{119}\right ) e^{6} b^{4} d}{3}-A \,b^{5} e^{7} x^{5}\right ) \sqrt {x \left (c x +b \right )}\right )}{512 \sqrt {d \left (b e -c d \right )}\, \left (e x +d \right )^{6} \left (b e -c d \right )^{4} d^{4}}\) | \(598\) |
| default | \(\text {Expression too large to display}\) | \(13664\) |
-7/512/(d*(b*e-c*d))^(1/2)*((12/7*(2*A*c^2-B*b*c)*d^2-24/7*(A*c-5/24*B*b)* e*b*d+A*b^2*e^2)*(e*x+d)^6*b^4*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^ (1/2))+(d*(b*e-c*d))^(1/2)*(24/7*c*(-1/2*b^4*B+c*(1/3*B*x+A)*b^3-2/3*c^2*x *(2/5*B*x+A)*b^2-8*c^3*x^2*(11/15*B*x+A)*b-16/3*c^4*(4/5*B*x+A)*x^3)*d^7-2 4/7*(-5/24*B*b^5+c*(107/36*B*x+A)*b^4-19/3*c^2*x*(89/285*B*x+A)*b^3-448/15 *c^3*(171/224*B*x+A)*x^2*b^2-72/5*c^4*x^3*(80/81*B*x+A)*b+32/15*c^5*x^4*(1 /3*B*x+A))*e*d^6+((85/21*B*x+A)*b^5-422/21*c*x*(1328/1055*B*x+A)*b^4-2456/ 21*c^2*(285/307*B*x+A)*x^2*b^3-816/35*c^3*x^3*(332/153*B*x+A)*b^2+1984/105 *c^4*(13/31*B*x+A)*x^4*b-128/105*A*c^5*x^5)*e^2*d^5+17/3*((198/119*B*x+A)* b^4+7144/595*c*(69/94*B*x+A)*x*b^3-2232/595*c^2*x^2*(-37/93*B*x+A)*b^2-736 /595*c^3*(51/46*B*x+A)*x^3*b+64/119*A*c^4*x^4)*x*e^3*b*d^4-562/35*x^2*((16 5/281*B*x+A)*b^3-642/281*c*(185/963*B*x+A)*x*b^2+168/281*c^2*x^2*(-5/63*B* x+A)*b+16/281*A*c^3*x^3)*e^4*b^2*d^3-66/5*x^3*((425/1386*B*x+A)*b^2-824/69 3*c*(65/824*B*x+A)*x*b+8/63*A*c^2*x^2)*e^5*b^3*d^2-17/3*x^4*((15/119*B*x+A )*b-58/119*A*c*x)*e^6*b^4*d-A*b^5*e^7*x^5)*(x*(c*x+b))^(1/2))/(e*x+d)^6/(b *e-c*d)^4/d^4
Leaf count of result is larger than twice the leaf count of optimal. 1861 vs. \(2 (371) = 742\).
Time = 1.08 (sec) , antiderivative size = 3734, normalized size of antiderivative = 9.29 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]
[1/15360*(15*(7*A*b^6*d^6*e^2 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^8 + (5*B*b^6 - 24*A*b^5*c)*d^7*e + (7*A*b^6*e^8 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^2*e^6 + (5*B*b^6 - 24*A*b^5*c)*d*e^7)*x^6 + 6*(7*A*b^6*d*e^7 - 12*(B*b^5*c - 2*A*b ^4*c^2)*d^3*e^5 + (5*B*b^6 - 24*A*b^5*c)*d^2*e^6)*x^5 + 15*(7*A*b^6*d^2*e^ 6 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^4*e^4 + (5*B*b^6 - 24*A*b^5*c)*d^3*e^5)*x ^4 + 20*(7*A*b^6*d^3*e^5 - 12*(B*b^5*c - 2*A*b^4*c^2)*d^5*e^3 + (5*B*b^6 - 24*A*b^5*c)*d^4*e^4)*x^3 + 15*(7*A*b^6*d^4*e^4 - 12*(B*b^5*c - 2*A*b^4*c^ 2)*d^6*e^2 + (5*B*b^6 - 24*A*b^5*c)*d^5*e^3)*x^2 + 6*(7*A*b^6*d^5*e^3 - 12 *(B*b^5*c - 2*A*b^4*c^2)*d^7*e + (5*B*b^6 - 24*A*b^5*c)*d^6*e^2)*x)*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*(105*A*b^6*d^6*e^3 + 180*(B*b^4*c^2 - 2*A*b^3*c^3 )*d^9 - 15*(17*B*b^5*c - 48*A*b^4*c^2)*d^8*e + 15*(5*B*b^6 - 31*A*b^5*c)*d ^7*e^2 + (256*B*c^6*d^8*e - 105*A*b^6*d*e^8 - 64*(17*B*b*c^5 - 2*A*c^6)*d^ 7*e^2 + 16*(103*B*b^2*c^4 - 28*A*b*c^5)*d^6*e^3 - 32*(28*B*b^3*c^3 - 13*A* b^2*c^4)*d^5*e^4 - 10*(5*B*b^4*c^2 - 8*A*b^3*c^3)*d^4*e^5 + (205*B*b^5*c - 466*A*b^4*c^2)*d^3*e^6 - 5*(15*B*b^6 - 79*A*b^5*c)*d^2*e^7)*x^5 + (1536*B *c^6*d^9 - 595*A*b^6*d^2*e^7 - 256*(26*B*b*c^5 - 3*A*c^6)*d^8*e + 64*(163* B*b^2*c^4 - 43*A*b*c^5)*d^7*e^2 - 40*(155*B*b^3*c^3 - 68*A*b^2*c^4)*d^6*e^ 3 + 4*(37*B*b^4*c^2 + 68*A*b^3*c^3)*d^5*e^4 + (1165*B*b^5*c - 2656*A*b^4*c ^2)*d^4*e^5 - (425*B*b^6 - 2243*A*b^5*c)*d^3*e^6)*x^4 - 6*(231*A*b^6*d^...
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{7}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 5881 vs. \(2 (371) = 742\).
Time = 0.41 (sec) , antiderivative size = 5881, normalized size of antiderivative = 14.63 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\text {Too large to display} \]
-1/512*(12*B*b^5*c*d^2 - 24*A*b^4*c^2*d^2 - 5*B*b^6*d*e + 24*A*b^5*c*d*e - 7*A*b^6*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt (-c*d^2 + b*d*e))/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c* d^5*e^3 + b^4*d^4*e^4)*sqrt(-c*d^2 + b*d*e)) + 1/7680*(180*(sqrt(c)*x - sq rt(c*x^2 + b*x))^11*B*b^5*c*d^2*e^10 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x)) ^11*A*b^4*c^2*d^2*e^10 - 75*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*B*b^6*d*e^1 1 + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^5*c*d*e^11 - 105*(sqrt(c)*x - sqrt(c*x^2 + b*x))^11*A*b^6*e^12 + 15360*(sqrt(c)*x - sqrt(c*x^2 + b*x) )^10*B*c^(13/2)*d^8*e^4 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b*c^( 11/2)*d^7*e^5 + 92160*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^2*c^(9/2)*d^6 *e^6 - 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^3*c^(7/2)*d^5*e^7 + 15 360*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*B*b^4*c^(5/2)*d^4*e^8 + 1980*(sqrt( c)*x - sqrt(c*x^2 + b*x))^10*B*b^5*c^(3/2)*d^3*e^9 - 3960*(sqrt(c)*x - sqr t(c*x^2 + b*x))^10*A*b^4*c^(5/2)*d^3*e^9 - 825*(sqrt(c)*x - sqrt(c*x^2 + b *x))^10*B*b^6*sqrt(c)*d^2*e^10 + 3960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A *b^5*c^(3/2)*d^2*e^10 - 1155*(sqrt(c)*x - sqrt(c*x^2 + b*x))^10*A*b^6*sqrt (c)*d*e^11 + 40960*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^7*d^9*e^3 - 11776 0*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*b*c^6*d^8*e^4 + 20480*(sqrt(c)*x - s qrt(c*x^2 + b*x))^9*A*c^7*d^8*e^4 + 61440*(sqrt(c)*x - sqrt(c*x^2 + b*x))^ 9*B*b^2*c^5*d^7*e^5 - 81920*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*A*b*c^6*d...
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^7} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^7} \,d x \]