Integrand size = 22, antiderivative size = 377 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=-\frac {5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^6 d}-\frac {5 \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{96 b^5 d (b c-a d)}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}-\frac {2 (6 b c-11 a d) x^2 (c+d x)^{5/2}}{3 b^2 (b c-a d) \sqrt {a+b x}}+\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2-156 a b c d+231 a^2 d^2+2 b d (59 b c-99 a d) x\right )}{24 b^4 d (b c-a d)}-\frac {5 (b c-a d) \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{13/2} d^{3/2}} \]
-2/3*x^3*(d*x+c)^(5/2)/b/(b*x+a)^(3/2)-5/64*(-a*d+b*c)*(231*a^3*d^3-189*a^ 2*b*c*d^2+21*a*b^2*c^2*d+b^3*c^3)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d *x+c)^(1/2))/b^(13/2)/d^(3/2)-2/3*(-11*a*d+6*b*c)*x^2*(d*x+c)^(5/2)/b^2/(- a*d+b*c)/(b*x+a)^(1/2)-5/96*(231*a^3*d^3-189*a^2*b*c*d^2+21*a*b^2*c^2*d+b^ 3*c^3)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^5/d/(-a*d+b*c)+1/24*(d*x+c)^(5/2)*(5* b^2*c^2-156*a*b*c*d+231*a^2*d^2+2*b*d*(-99*a*d+59*b*c)*x)*(b*x+a)^(1/2)/b^ 4/d/(-a*d+b*c)-5/64*(231*a^3*d^3-189*a^2*b*c*d^2+21*a*b^2*c^2*d+b^3*c^3)*( b*x+a)^(1/2)*(d*x+c)^(1/2)/b^6/d
Time = 10.97 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {\sqrt {d} \left (-3465 a^5 d^3+105 a^4 b d^2 (49 c-44 d x)-21 a^3 b^2 d \left (83 c^2-334 c d x+33 d^2 x^2\right )+b^5 x^2 \left (15 c^3+118 c^2 d x+136 c d^2 x^2+48 d^3 x^3\right )+3 a^2 b^3 \left (5 c^3-824 c^2 d x+387 c d^2 x^2+66 d^3 x^3\right )-a b^4 x \left (-30 c^3+483 c^2 d x+316 c d^2 x^2+88 d^3 x^3\right )\right )}{(a+b x)^{3/2}}-\frac {15 \sqrt {b c-a d} \left (b^3 c^3+21 a b^2 c^2 d-189 a^2 b c d^2+231 a^3 d^3\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{192 b^6 d^{3/2}} \]
(Sqrt[c + d*x]*((Sqrt[d]*(-3465*a^5*d^3 + 105*a^4*b*d^2*(49*c - 44*d*x) - 21*a^3*b^2*d*(83*c^2 - 334*c*d*x + 33*d^2*x^2) + b^5*x^2*(15*c^3 + 118*c^2 *d*x + 136*c*d^2*x^2 + 48*d^3*x^3) + 3*a^2*b^3*(5*c^3 - 824*c^2*d*x + 387* c*d^2*x^2 + 66*d^3*x^3) - a*b^4*x*(-30*c^3 + 483*c^2*d*x + 316*c*d^2*x^2 + 88*d^3*x^3)))/(a + b*x)^(3/2) - (15*Sqrt[b*c - a*d]*(b^3*c^3 + 21*a*b^2*c ^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^3)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt [b*c - a*d]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(192*b^6*d^(3/2))
Time = 0.40 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {108, 27, 167, 27, 164, 60, 60, 66, 221}
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.
| \(\displaystyle \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 \int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{2 (a+b x)^{3/2}}dx}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{(a+b x)^{3/2}}dx}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\frac {2 \int \frac {x (c+d x)^{3/2} (4 c (6 b c-11 a d)+d (59 b c-99 a d) x)}{2 \sqrt {a+b x}}dx}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x (c+d x)^{3/2} (4 c (6 b c-11 a d)+d (59 b c-99 a d) x)}{\sqrt {a+b x}}dx}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{8 b^2 d}-\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{16 b^2 d}}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{8 b^2 d}-\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d}}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{8 b^2 d}-\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d}}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{8 b^2 d}-\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d}}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (231 a^2 d^2+2 b d x (59 b c-99 a d)-156 a b c d+5 b^2 c^2\right )}{8 b^2 d}-\frac {5 \left (231 a^3 d^3-189 a^2 b c d^2+21 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d}}{b (b c-a d)}-\frac {2 x^2 (c+d x)^{5/2} (6 b c-11 a d)}{b \sqrt {a+b x} (b c-a d)}}{3 b}-\frac {2 x^3 (c+d x)^{5/2}}{3 b (a+b x)^{3/2}}\) |
(-2*x^3*(c + d*x)^(5/2))/(3*b*(a + b*x)^(3/2)) + ((-2*(6*b*c - 11*a*d)*x^2 *(c + d*x)^(5/2))/(b*(b*c - a*d)*Sqrt[a + b*x]) + ((Sqrt[a + b*x]*(c + d*x )^(5/2)*(5*b^2*c^2 - 156*a*b*c*d + 231*a^2*d^2 + 2*b*d*(59*b*c - 99*a*d)*x ))/(8*b^2*d) - (5*(b^3*c^3 + 21*a*b^2*c^2*d - 189*a^2*b*c*d^2 + 231*a^3*d^ 3)*((Sqrt[a + b*x]*(c + d*x)^(3/2))/(2*b) + (3*(b*c - a*d)*((Sqrt[a + b*x] *Sqrt[c + d*x])/b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]* Sqrt[c + d*x])])/(b^(3/2)*Sqrt[d])))/(4*b)))/(16*b^2*d))/(b*(b*c - a*d)))/ (3*b)
3.8.94.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Leaf count of result is larger than twice the leaf count of optimal. \(1365\) vs. \(2(333)=666\).
Time = 1.72 (sec) , antiderivative size = 1366, normalized size of antiderivative = 3.62
1/384*(d*x+c)^(1/2)*(-300*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^ (1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^5*c^3*d*x^2-12600*ln(1/2*(2*b*d*x+2*((b*x+ a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*c*d^3*x+6300*l n(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2)) *a^3*b^3*c^2*d^2*x-1386*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^2*d^3*x^ 2-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^ (1/2))*b^6*c^4*x^2-15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2 )+a*d+b*c)/(b*d)^(1/2))*a^2*b^4*c^4-6930*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/ 2)*a^5*d^3-176*a*b^4*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+272*b^5*c *d^2*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3465*ln(1/2*(2*b*d*x+2*((b*x+ a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b^2*d^4*x^2-9240*( (b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b*d^3*x+10290*((b*x+a)*(d*x+c))^(1/ 2)*(b*d)^(1/2)*a^4*b*c*d^2+2322*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^ 3*c*d^2*x^2+396*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^3*d^3*x^3+236*(( b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^5*c^2*d*x^3+14028*((b*x+a)*(d*x+c))^(1 /2)*(b*d)^(1/2)*a^3*b^2*c*d^2*x+3465*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^( 1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^6*d^4+6930*ln(1/2*(2*b*d*x+2*((b* x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*b*d^4*x-6300*ln( 1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a ^5*b*c*d^3+3150*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a...
Time = 0.65 (sec) , antiderivative size = 1060, normalized size of antiderivative = 2.81 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (a^{2} b^{4} c^{4} + 20 \, a^{3} b^{3} c^{3} d - 210 \, a^{4} b^{2} c^{2} d^{2} + 420 \, a^{5} b c d^{3} - 231 \, a^{6} d^{4} + {\left (b^{6} c^{4} + 20 \, a b^{5} c^{3} d - 210 \, a^{2} b^{4} c^{2} d^{2} + 420 \, a^{3} b^{3} c d^{3} - 231 \, a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} + 20 \, a^{2} b^{4} c^{3} d - 210 \, a^{3} b^{3} c^{2} d^{2} + 420 \, a^{4} b^{2} c d^{3} - 231 \, a^{5} b d^{4}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (48 \, b^{6} d^{4} x^{5} + 15 \, a^{2} b^{4} c^{3} d - 1743 \, a^{3} b^{3} c^{2} d^{2} + 5145 \, a^{4} b^{2} c d^{3} - 3465 \, a^{5} b d^{4} + 8 \, {\left (17 \, b^{6} c d^{3} - 11 \, a b^{5} d^{4}\right )} x^{4} + 2 \, {\left (59 \, b^{6} c^{2} d^{2} - 158 \, a b^{5} c d^{3} + 99 \, a^{2} b^{4} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{6} c^{3} d - 161 \, a b^{5} c^{2} d^{2} + 387 \, a^{2} b^{4} c d^{3} - 231 \, a^{3} b^{3} d^{4}\right )} x^{2} + 6 \, {\left (5 \, a b^{5} c^{3} d - 412 \, a^{2} b^{4} c^{2} d^{2} + 1169 \, a^{3} b^{3} c d^{3} - 770 \, a^{4} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, {\left (b^{9} d^{2} x^{2} + 2 \, a b^{8} d^{2} x + a^{2} b^{7} d^{2}\right )}}, \frac {15 \, {\left (a^{2} b^{4} c^{4} + 20 \, a^{3} b^{3} c^{3} d - 210 \, a^{4} b^{2} c^{2} d^{2} + 420 \, a^{5} b c d^{3} - 231 \, a^{6} d^{4} + {\left (b^{6} c^{4} + 20 \, a b^{5} c^{3} d - 210 \, a^{2} b^{4} c^{2} d^{2} + 420 \, a^{3} b^{3} c d^{3} - 231 \, a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} + 20 \, a^{2} b^{4} c^{3} d - 210 \, a^{3} b^{3} c^{2} d^{2} + 420 \, a^{4} b^{2} c d^{3} - 231 \, a^{5} b d^{4}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) + 2 \, {\left (48 \, b^{6} d^{4} x^{5} + 15 \, a^{2} b^{4} c^{3} d - 1743 \, a^{3} b^{3} c^{2} d^{2} + 5145 \, a^{4} b^{2} c d^{3} - 3465 \, a^{5} b d^{4} + 8 \, {\left (17 \, b^{6} c d^{3} - 11 \, a b^{5} d^{4}\right )} x^{4} + 2 \, {\left (59 \, b^{6} c^{2} d^{2} - 158 \, a b^{5} c d^{3} + 99 \, a^{2} b^{4} d^{4}\right )} x^{3} + 3 \, {\left (5 \, b^{6} c^{3} d - 161 \, a b^{5} c^{2} d^{2} + 387 \, a^{2} b^{4} c d^{3} - 231 \, a^{3} b^{3} d^{4}\right )} x^{2} + 6 \, {\left (5 \, a b^{5} c^{3} d - 412 \, a^{2} b^{4} c^{2} d^{2} + 1169 \, a^{3} b^{3} c d^{3} - 770 \, a^{4} b^{2} d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, {\left (b^{9} d^{2} x^{2} + 2 \, a b^{8} d^{2} x + a^{2} b^{7} d^{2}\right )}}\right ] \]
[-1/768*(15*(a^2*b^4*c^4 + 20*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^ 5*b*c*d^3 - 231*a^6*d^4 + (b^6*c^4 + 20*a*b^5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c*d^3 - 231*a^4*b^2*d^4)*x^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c^3 *d - 210*a^3*b^3*c^2*d^2 + 420*a^4*b^2*c*d^3 - 231*a^5*b*d^4)*x)*sqrt(b*d) *log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a* d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(4 8*b^6*d^4*x^5 + 15*a^2*b^4*c^3*d - 1743*a^3*b^3*c^2*d^2 + 5145*a^4*b^2*c*d ^3 - 3465*a^5*b*d^4 + 8*(17*b^6*c*d^3 - 11*a*b^5*d^4)*x^4 + 2*(59*b^6*c^2* d^2 - 158*a*b^5*c*d^3 + 99*a^2*b^4*d^4)*x^3 + 3*(5*b^6*c^3*d - 161*a*b^5*c ^2*d^2 + 387*a^2*b^4*c*d^3 - 231*a^3*b^3*d^4)*x^2 + 6*(5*a*b^5*c^3*d - 412 *a^2*b^4*c^2*d^2 + 1169*a^3*b^3*c*d^3 - 770*a^4*b^2*d^4)*x)*sqrt(b*x + a)* sqrt(d*x + c))/(b^9*d^2*x^2 + 2*a*b^8*d^2*x + a^2*b^7*d^2), 1/384*(15*(a^2 *b^4*c^4 + 20*a^3*b^3*c^3*d - 210*a^4*b^2*c^2*d^2 + 420*a^5*b*c*d^3 - 231* a^6*d^4 + (b^6*c^4 + 20*a*b^5*c^3*d - 210*a^2*b^4*c^2*d^2 + 420*a^3*b^3*c* d^3 - 231*a^4*b^2*d^4)*x^2 + 2*(a*b^5*c^4 + 20*a^2*b^4*c^3*d - 210*a^3*b^3 *c^2*d^2 + 420*a^4*b^2*c*d^3 - 231*a^5*b*d^4)*x)*sqrt(-b*d)*arctan(1/2*(2* b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a *b*c*d + (b^2*c*d + a*b*d^2)*x)) + 2*(48*b^6*d^4*x^5 + 15*a^2*b^4*c^3*d - 1743*a^3*b^3*c^2*d^2 + 5145*a^4*b^2*c*d^3 - 3465*a^5*b*d^4 + 8*(17*b^6*c*d ^3 - 11*a*b^5*d^4)*x^4 + 2*(59*b^6*c^2*d^2 - 158*a*b^5*c*d^3 + 99*a^2*b...
\[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^{3} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 943 vs. \(2 (333) = 666\).
Time = 0.65 (sec) , antiderivative size = 943, normalized size of antiderivative = 2.50 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {1}{192} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{8}} + \frac {17 \, b^{32} c d^{7} {\left | b \right |} - 41 \, a b^{31} d^{8} {\left | b \right |}}{b^{39} d^{6}}\right )} + \frac {59 \, b^{33} c^{2} d^{6} {\left | b \right |} - 430 \, a b^{32} c d^{7} {\left | b \right |} + 515 \, a^{2} b^{31} d^{8} {\left | b \right |}}{b^{39} d^{6}}\right )} + \frac {3 \, {\left (5 \, b^{34} c^{3} d^{5} {\left | b \right |} - 279 \, a b^{33} c^{2} d^{6} {\left | b \right |} + 975 \, a^{2} b^{32} c d^{7} {\left | b \right |} - 765 \, a^{3} b^{31} d^{8} {\left | b \right |}\right )}}{b^{39} d^{6}}\right )} \sqrt {b x + a} + \frac {5 \, {\left (b^{4} c^{4} {\left | b \right |} + 20 \, a b^{3} c^{3} d {\left | b \right |} - 210 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 420 \, a^{3} b c d^{3} {\left | b \right |} - 231 \, a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, \sqrt {b d} b^{7} d} - \frac {4 \, {\left (9 \, a^{2} b^{7} c^{5} d {\left | b \right |} - 52 \, a^{3} b^{6} c^{4} d^{2} {\left | b \right |} + 118 \, a^{4} b^{5} c^{3} d^{3} {\left | b \right |} - 132 \, a^{5} b^{4} c^{2} d^{4} {\left | b \right |} + 73 \, a^{6} b^{3} c d^{5} {\left | b \right |} - 16 \, a^{7} b^{2} d^{6} {\left | b \right |} - 18 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{5} c^{4} d {\left | b \right |} + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{4} c^{3} d^{2} {\left | b \right |} - 144 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{3} c^{2} d^{3} {\left | b \right |} + 108 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b^{2} c d^{4} {\left | b \right |} - 30 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{6} b d^{5} {\left | b \right |} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{3} c^{3} d {\left | b \right |} - 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b^{2} c^{2} d^{2} {\left | b \right |} + 45 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} b c d^{3} {\left | b \right |} - 18 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{5} d^{4} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} \sqrt {b d} b^{6}} \]
1/192*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(6*(b* x + a)*d^2*abs(b)/b^8 + (17*b^32*c*d^7*abs(b) - 41*a*b^31*d^8*abs(b))/(b^3 9*d^6)) + (59*b^33*c^2*d^6*abs(b) - 430*a*b^32*c*d^7*abs(b) + 515*a^2*b^31 *d^8*abs(b))/(b^39*d^6)) + 3*(5*b^34*c^3*d^5*abs(b) - 279*a*b^33*c^2*d^6*a bs(b) + 975*a^2*b^32*c*d^7*abs(b) - 765*a^3*b^31*d^8*abs(b))/(b^39*d^6))*s qrt(b*x + a) + 5/128*(b^4*c^4*abs(b) + 20*a*b^3*c^3*d*abs(b) - 210*a^2*b^2 *c^2*d^2*abs(b) + 420*a^3*b*c*d^3*abs(b) - 231*a^4*d^4*abs(b))*log((sqrt(b *d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^7 *d) - 4/3*(9*a^2*b^7*c^5*d*abs(b) - 52*a^3*b^6*c^4*d^2*abs(b) + 118*a^4*b^ 5*c^3*d^3*abs(b) - 132*a^5*b^4*c^2*d^4*abs(b) + 73*a^6*b^3*c*d^5*abs(b) - 16*a^7*b^2*d^6*abs(b) - 18*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^5*c^4*d*abs(b) + 84*(sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^4*c^3*d^2*abs(b) - 144*(sqrt(b* d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^3*c^2*d^3* abs(b) + 108*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d ))^2*a^5*b^2*c*d^4*abs(b) - 30*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b* x + a)*b*d - a*b*d))^2*a^6*b*d^5*abs(b) + 9*(sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b^3*c^3*d*abs(b) - 36*(sqrt(b*d)*s qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*b^2*c^2*d^2*abs( b) + 45*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))...
Timed out. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]