Integrand size = 22, antiderivative size = 346 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {3 (b c-a d) \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{128 b^6 d^2}+\frac {\left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{64 b^5 d^2}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b^2}-\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (5 b^2 c^2+30 a b c d-231 a^2 d^2-2 b d (5 b c-99 a d) x\right )}{80 b^4 d^2}+\frac {3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{128 b^{13/2} d^{5/2}} \]
3/128*(-a*d+b*c)^2*(-231*a^3*d^3+63*a^2*b*c*d^2+7*a*b^2*c^2*d+b^3*c^3)*arc tanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(13/2)/d^(5/2)-2*x^3*( d*x+c)^(5/2)/b/(b*x+a)^(1/2)+1/64*(-231*a^3*d^3+63*a^2*b*c*d^2+7*a*b^2*c^2 *d+b^3*c^3)*(d*x+c)^(3/2)*(b*x+a)^(1/2)/b^5/d^2+11/5*x^2*(d*x+c)^(5/2)*(b* x+a)^(1/2)/b^2-1/80*(d*x+c)^(5/2)*(5*b^2*c^2+30*a*b*c*d-231*a^2*d^2-2*b*d* (-99*a*d+5*b*c)*x)*(b*x+a)^(1/2)/b^4/d^2+3/128*(-a*d+b*c)*(-231*a^3*d^3+63 *a^2*b*c*d^2+7*a*b^2*c^2*d+b^3*c^3)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^6/d^2
Time = 0.90 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (3465 a^5 d^4+105 a^4 b d^3 (-64 c+11 d x)-42 a^3 b^2 d^2 \left (-79 c^2+57 c d x+11 d^2 x^2\right )+2 a^2 b^3 d \left (-40 c^3+662 c^2 d x+459 c d^2 x^2+132 d^3 x^3\right )+b^5 x \left (-15 c^4+10 c^3 d x+248 c^2 d^2 x^2+336 c d^3 x^3+128 d^4 x^4\right )-a b^4 \left (15 c^4+70 c^3 d x+466 c^2 d^2 x^2+512 c d^3 x^3+176 d^4 x^4\right )\right )}{640 b^6 d^2 \sqrt {a+b x}}+\frac {3 (b c-a d)^2 \left (b^3 c^3+7 a b^2 c^2 d+63 a^2 b c d^2-231 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{128 b^{13/2} d^{5/2}} \]
(Sqrt[c + d*x]*(3465*a^5*d^4 + 105*a^4*b*d^3*(-64*c + 11*d*x) - 42*a^3*b^2 *d^2*(-79*c^2 + 57*c*d*x + 11*d^2*x^2) + 2*a^2*b^3*d*(-40*c^3 + 662*c^2*d* x + 459*c*d^2*x^2 + 132*d^3*x^3) + b^5*x*(-15*c^4 + 10*c^3*d*x + 248*c^2*d ^2*x^2 + 336*c*d^3*x^3 + 128*d^4*x^4) - a*b^4*(15*c^4 + 70*c^3*d*x + 466*c ^2*d^2*x^2 + 512*c*d^3*x^3 + 176*d^4*x^4)))/(640*b^6*d^2*Sqrt[a + b*x]) + (3*(b*c - a*d)^2*(b^3*c^3 + 7*a*b^2*c^2*d + 63*a^2*b*c*d^2 - 231*a^3*d^3)* ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(128*b^(13/2)*d^ (5/2))
Time = 0.36 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {108, 27, 170, 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.
| \(\displaystyle \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2 \int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{2 \sqrt {a+b x}}dx}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 (c+d x)^{3/2} (6 c+11 d x)}{\sqrt {a+b x}}dx}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {\frac {\int -\frac {d x (c+d x)^{3/2} (44 a c-(5 b c-99 a d) x)}{2 \sqrt {a+b x}}dx}{5 b d}+\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\int \frac {x (c+d x)^{3/2} (44 a c-(5 b c-99 a d) x)}{\sqrt {a+b x}}dx}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{8 b^2 d^2}-\frac {5 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}}dx}{16 b^2 d^2}}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{8 b^2 d^2}-\frac {5 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{8 b^2 d^2}-\frac {5 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{8 b^2 d^2}-\frac {5 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {11 x^2 \sqrt {a+b x} (c+d x)^{5/2}}{5 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (-231 a^2 d^2-2 b d x (5 b c-99 a d)+30 a b c d+5 b^2 c^2\right )}{8 b^2 d^2}-\frac {5 \left (-231 a^3 d^3+63 a^2 b c d^2+7 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{4 b}+\frac {\sqrt {a+b x} (c+d x)^{3/2}}{2 b}\right )}{16 b^2 d^2}}{10 b}}{b}-\frac {2 x^3 (c+d x)^{5/2}}{b \sqrt {a+b x}}\) |
(-2*x^3*(c + d*x)^(5/2))/(b*Sqrt[a + b*x]) + ((11*x^2*Sqrt[a + b*x]*(c + d *x)^(5/2))/(5*b) - ((Sqrt[a + b*x]*(c + d*x)^(5/2)*(5*b^2*c^2 + 30*a*b*c*d - 231*a^2*d^2 - 2*b*d*(5*b*c - 99*a*d)*x))/(8*b^2*d^2) - (5*(b^3*c^3 + 7* a*b^2*c^2*d + 63*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^2))/(10*b))/b
3.8.67.3.1 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
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. \(1264\) vs. \(2(304)=608\).
Time = 0.59 (sec) , antiderivative size = 1265, normalized size of antiderivative = 3.66
-1/1280*(d*x+c)^(1/2)*(352*a*b^4*d^4*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/ 2)-672*b^5*c*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-528*a^2*b^3*d^4*x ^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-496*b^5*c^2*d^2*x^3*((b*x+a)*(d*x+c ))^(1/2)*(b*d)^(1/2)+924*a^3*b^2*d^4*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/ 2)-20*b^5*c^3*d*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-2310*a^4*b*d^4*x*( (b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+932*a*b^4*c^2*d^2*x^2*((b*x+a)*(d*x+c)) ^(1/2)*(b*d)^(1/2)+4788*a^3*b^2*c*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2 )-2648*a^2*b^3*c^2*d^2*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+140*a*b^4*c^3 *d*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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*b^5*c^5-15*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^5*x-693 0*a^5*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-256*b^5*d^4*x^5*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)+30*b^5*c^4*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3 0*a*b^4*c^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^5*b*d^5*x-7875*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^4+5250*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^2*d^3-750*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^3*b^3*c^3*d^2-75*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...
Time = 0.51 (sec) , antiderivative size = 1008, normalized size of antiderivative = 2.91 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d + 50 \, a^{3} b^{3} c^{3} d^{2} - 350 \, a^{4} b^{2} c^{2} d^{3} + 525 \, a^{5} b c d^{4} - 231 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d + 50 \, a^{2} b^{4} c^{3} d^{2} - 350 \, a^{3} b^{3} c^{2} d^{3} + 525 \, a^{4} b^{2} c d^{4} - 231 \, a^{5} b d^{5}\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 (128 \, b^{6} d^{5} x^{5} - 15 \, a b^{5} c^{4} d - 80 \, a^{2} b^{4} c^{3} d^{2} + 3318 \, a^{3} b^{3} c^{2} d^{3} - 6720 \, a^{4} b^{2} c d^{4} + 3465 \, a^{5} b d^{5} + 16 \, {\left (21 \, b^{6} c d^{4} - 11 \, a b^{5} d^{5}\right )} x^{4} + 8 \, {\left (31 \, b^{6} c^{2} d^{3} - 64 \, a b^{5} c d^{4} + 33 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{6} c^{3} d^{2} - 233 \, a b^{5} c^{2} d^{3} + 459 \, a^{2} b^{4} c d^{4} - 231 \, a^{3} b^{3} d^{5}\right )} x^{2} - {\left (15 \, b^{6} c^{4} d + 70 \, a b^{5} c^{3} d^{2} - 1324 \, a^{2} b^{4} c^{2} d^{3} + 2394 \, a^{3} b^{3} c d^{4} - 1155 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{2560 \, {\left (b^{8} d^{3} x + a b^{7} d^{3}\right )}}, -\frac {15 \, {\left (a b^{5} c^{5} + 5 \, a^{2} b^{4} c^{4} d + 50 \, a^{3} b^{3} c^{3} d^{2} - 350 \, a^{4} b^{2} c^{2} d^{3} + 525 \, a^{5} b c d^{4} - 231 \, a^{6} d^{5} + {\left (b^{6} c^{5} + 5 \, a b^{5} c^{4} d + 50 \, a^{2} b^{4} c^{3} d^{2} - 350 \, a^{3} b^{3} c^{2} d^{3} + 525 \, a^{4} b^{2} c d^{4} - 231 \, a^{5} b d^{5}\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 (128 \, b^{6} d^{5} x^{5} - 15 \, a b^{5} c^{4} d - 80 \, a^{2} b^{4} c^{3} d^{2} + 3318 \, a^{3} b^{3} c^{2} d^{3} - 6720 \, a^{4} b^{2} c d^{4} + 3465 \, a^{5} b d^{5} + 16 \, {\left (21 \, b^{6} c d^{4} - 11 \, a b^{5} d^{5}\right )} x^{4} + 8 \, {\left (31 \, b^{6} c^{2} d^{3} - 64 \, a b^{5} c d^{4} + 33 \, a^{2} b^{4} d^{5}\right )} x^{3} + 2 \, {\left (5 \, b^{6} c^{3} d^{2} - 233 \, a b^{5} c^{2} d^{3} + 459 \, a^{2} b^{4} c d^{4} - 231 \, a^{3} b^{3} d^{5}\right )} x^{2} - {\left (15 \, b^{6} c^{4} d + 70 \, a b^{5} c^{3} d^{2} - 1324 \, a^{2} b^{4} c^{2} d^{3} + 2394 \, a^{3} b^{3} c d^{4} - 1155 \, a^{4} b^{2} d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{1280 \, {\left (b^{8} d^{3} x + a b^{7} d^{3}\right )}}\right ] \]
[-1/2560*(15*(a*b^5*c^5 + 5*a^2*b^4*c^4*d + 50*a^3*b^3*c^3*d^2 - 350*a^4*b ^2*c^2*d^3 + 525*a^5*b*c*d^4 - 231*a^6*d^5 + (b^6*c^5 + 5*a*b^5*c^4*d + 50 *a^2*b^4*c^3*d^2 - 350*a^3*b^3*c^2*d^3 + 525*a^4*b^2*c*d^4 - 231*a^5*b*d^5 )*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*(128*b^6*d^5*x^5 - 15*a*b^5*c^4*d - 80*a^2*b^4*c^3*d^2 + 3318* a^3*b^3*c^2*d^3 - 6720*a^4*b^2*c*d^4 + 3465*a^5*b*d^5 + 16*(21*b^6*c*d^4 - 11*a*b^5*d^5)*x^4 + 8*(31*b^6*c^2*d^3 - 64*a*b^5*c*d^4 + 33*a^2*b^4*d^5)* x^3 + 2*(5*b^6*c^3*d^2 - 233*a*b^5*c^2*d^3 + 459*a^2*b^4*c*d^4 - 231*a^3*b ^3*d^5)*x^2 - (15*b^6*c^4*d + 70*a*b^5*c^3*d^2 - 1324*a^2*b^4*c^2*d^3 + 23 94*a^3*b^3*c*d^4 - 1155*a^4*b^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^8* d^3*x + a*b^7*d^3), -1/1280*(15*(a*b^5*c^5 + 5*a^2*b^4*c^4*d + 50*a^3*b^3* c^3*d^2 - 350*a^4*b^2*c^2*d^3 + 525*a^5*b*c*d^4 - 231*a^6*d^5 + (b^6*c^5 + 5*a*b^5*c^4*d + 50*a^2*b^4*c^3*d^2 - 350*a^3*b^3*c^2*d^3 + 525*a^4*b^2*c* d^4 - 231*a^5*b*d^5)*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*(128*b^6*d^5*x^5 - 15*a*b^5*c^4*d - 80*a^2*b^4*c^3*d^2 + 3318* a^3*b^3*c^2*d^3 - 6720*a^4*b^2*c*d^4 + 3465*a^5*b*d^5 + 16*(21*b^6*c*d^4 - 11*a*b^5*d^5)*x^4 + 8*(31*b^6*c^2*d^3 - 64*a*b^5*c*d^4 + 33*a^2*b^4*d^5)* x^3 + 2*(5*b^6*c^3*d^2 - 233*a*b^5*c^2*d^3 + 459*a^2*b^4*c*d^4 - 231*a^...
\[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^{3} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Time = 0.59 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.54 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{640} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (4 \, {\left (b x + a\right )} {\left (2 \, {\left (b x + a\right )} {\left (\frac {8 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{8}} + \frac {3 \, {\left (7 \, b^{40} c d^{9} {\left | b \right |} - 17 \, a b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} + \frac {31 \, b^{41} c^{2} d^{8} {\left | b \right |} - 232 \, a b^{40} c d^{9} {\left | b \right |} + 281 \, a^{2} b^{39} d^{10} {\left | b \right |}}{b^{47} d^{8}}\right )} + \frac {5 \, {\left (b^{42} c^{3} d^{7} {\left | b \right |} - 121 \, a b^{41} c^{2} d^{8} {\left | b \right |} + 447 \, a^{2} b^{40} c d^{9} {\left | b \right |} - 359 \, a^{3} b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (b^{43} c^{4} d^{6} {\left | b \right |} + 6 \, a b^{42} c^{3} d^{7} {\left | b \right |} - 200 \, a^{2} b^{41} c^{2} d^{8} {\left | b \right |} + 474 \, a^{3} b^{40} c d^{9} {\left | b \right |} - 281 \, a^{4} b^{39} d^{10} {\left | b \right |}\right )}}{b^{47} d^{8}}\right )} \sqrt {b x + a} + \frac {4 \, {\left (a^{3} b^{3} c^{3} d {\left | b \right |} - 3 \, a^{4} b^{2} c^{2} d^{2} {\left | b \right |} + 3 \, a^{5} b c d^{3} {\left | b \right |} - a^{6} d^{4} {\left | b \right |}\right )}}{{\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 )} \sqrt {b d} b^{6}} - \frac {3 \, {\left (b^{5} c^{5} {\left | b \right |} + 5 \, a b^{4} c^{4} d {\left | b \right |} + 50 \, a^{2} b^{3} c^{3} d^{2} {\left | b \right |} - 350 \, a^{3} b^{2} c^{2} d^{3} {\left | b \right |} + 525 \, a^{4} b c d^{4} {\left | b \right |} - 231 \, a^{5} d^{5} {\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 )}{256 \, \sqrt {b d} b^{7} d^{2}} \]
1/640*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(4*(b*x + a)*(2*(b*x + a)*(8* (b*x + a)*d^2*abs(b)/b^8 + 3*(7*b^40*c*d^9*abs(b) - 17*a*b^39*d^10*abs(b)) /(b^47*d^8)) + (31*b^41*c^2*d^8*abs(b) - 232*a*b^40*c*d^9*abs(b) + 281*a^2 *b^39*d^10*abs(b))/(b^47*d^8)) + 5*(b^42*c^3*d^7*abs(b) - 121*a*b^41*c^2*d ^8*abs(b) + 447*a^2*b^40*c*d^9*abs(b) - 359*a^3*b^39*d^10*abs(b))/(b^47*d^ 8))*(b*x + a) - 15*(b^43*c^4*d^6*abs(b) + 6*a*b^42*c^3*d^7*abs(b) - 200*a^ 2*b^41*c^2*d^8*abs(b) + 474*a^3*b^40*c*d^9*abs(b) - 281*a^4*b^39*d^10*abs( b))/(b^47*d^8))*sqrt(b*x + a) + 4*(a^3*b^3*c^3*d*abs(b) - 3*a^4*b^2*c^2*d^ 2*abs(b) + 3*a^5*b*c*d^3*abs(b) - a^6*d^4*abs(b))/((b^2*c - a*b*d - (sqrt( b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)*sqrt(b*d)*b^6 ) - 3/256*(b^5*c^5*abs(b) + 5*a*b^4*c^4*d*abs(b) + 50*a^2*b^3*c^3*d^2*abs( b) - 350*a^3*b^2*c^2*d^3*abs(b) + 525*a^4*b*c*d^4*abs(b) - 231*a^5*d^5*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^2)
Timed out. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^{3/2}} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{3/2}} \,d x \]