Integrand size = 22, antiderivative size = 258 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}} \]
1/4*(3*a^2*d^2+10*a*b*c*d+35*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2 )/(d*x+c)^(1/2))/b^(5/2)/d^(9/2)-2/3*c*x^3*(b*x+a)^(1/2)/d/(-a*d+b*c)/(d*x +c)^(3/2)-2/3*c*(-9*a*d+7*b*c)*x^2*(b*x+a)^(1/2)/d^2/(-a*d+b*c)^2/(d*x+c)^ (1/2)-1/12*(105*b^3*c^3-145*a*b^2*c^2*d+15*a^2*b*c*d^2+9*a^3*d^3-2*b*d*(3* a^2*d^2-46*a*b*c*d+35*b^2*c^2)*x)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/d^4/(-a* d+b*c)^2
Time = 0.48 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {a+b x} \left (9 a^3 d^3 (c+d x)^2+3 a^2 b d^2 (5 c-2 d x) (c+d x)^2-a b^2 c d \left (145 c^3+198 c^2 d x+33 c d^2 x^2-12 d^3 x^3\right )+b^3 c^2 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )}{12 b^2 d^4 (b c-a d)^2 (c+d x)^{3/2}}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}} \]
-1/12*(Sqrt[a + b*x]*(9*a^3*d^3*(c + d*x)^2 + 3*a^2*b*d^2*(5*c - 2*d*x)*(c + d*x)^2 - a*b^2*c*d*(145*c^3 + 198*c^2*d*x + 33*c*d^2*x^2 - 12*d^3*x^3) + b^3*c^2*(105*c^3 + 140*c^2*d*x + 21*c*d^2*x^2 - 6*d^3*x^3)))/(b^2*d^4*(b *c - a*d)^2*(c + d*x)^(3/2)) + ((35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcT anh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(9/2))
Time = 0.40 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {109, 27, 167, 27, 164, 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.
| \(\displaystyle \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \int \frac {x^2 (6 a c+(7 b c-3 a d) x)}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {x^2 (6 a c+(7 b c-3 a d) x)}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {x \left (4 a c (7 b c-9 a d)+\left (35 b^2 c^2-46 a b d c+3 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {x \left (4 a c (7 b c-9 a d)+\left (35 b^2 c^2-46 a b d c+3 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {\frac {\frac {3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{8 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {\frac {3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{4 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 (b c-a d)^2 \left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{4 b^2 d^2}}{d (b c-a d)}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{d \sqrt {c+d x} (b c-a d)}}{3 d (b c-a d)}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)}\) |
(-2*c*x^3*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) + ((-2*c*(7*b*c - 9*a*d)*x^2*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) + (-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x]*(105*b^3*c^3 - 145*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 9* a^3*d^3 - 2*b*d*(35*b^2*c^2 - 46*a*b*c*d + 3*a^2*d^2)*x))/(b^2*d^2) + (3*( b*c - a*d)^2*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(5/2)*d^(5/2)))/(d*(b*c - a*d)))/( 3*d*(b*c - a*d))
3.8.48.3.1 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[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[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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. \(1286\) vs. \(2(226)=452\).
Time = 0.58 (sec) , antiderivative size = 1287, normalized size of antiderivative = 4.99
1/24*(b*x+a)^(1/2)*(-36*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c*d^4*x+12 *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*c^3*d^3+54*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^2*c^4*d^2-180*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^3*c^5*d-48*((b*x+a)*(d*x+c ))^(1/2)*(b*d)^(1/2)*a^2*b*c^2*d^3*x+396*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/ 2)*a*b^2*c^3*d^2*x+9*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*d^6*x^2+9*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*c^2*d^4-210*((b*x+a)*(d*x+c))^( 1/2)*(b*d)^(1/2)*b^3*c^5+12*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*d^5* x^3+108*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^2*c^3*d^3*x-360*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^3*c^4*d^2*x+18*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*c*d^5*x+105*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 ^4*c^6-24*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^2*c*d^4*x^3-42*((b*x+a)* (d*x+c))^(1/2)*(b*d)^(1/2)*b^3*c^3*d^2*x^2+24*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*c^2*d^4*x+12*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 *c*d^5*x^2+54*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d...
Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (226) = 452\).
Time = 0.53 (sec) , antiderivative size = 1144, normalized size of antiderivative = 4.43 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} c^{6} - 60 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} + 4 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4} + {\left (35 \, b^{4} c^{4} d^{2} - 60 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} + 4 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{5} d - 60 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} + 4 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c 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 (105 \, b^{4} c^{5} d - 145 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} + 9 \, a^{3} b c^{2} d^{4} - 6 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x^{3} + 3 \, {\left (7 \, b^{4} c^{3} d^{3} - 11 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + 3 \, a^{3} b d^{6}\right )} x^{2} + 2 \, {\left (70 \, b^{4} c^{4} d^{2} - 99 \, a b^{3} c^{3} d^{3} + 12 \, a^{2} b^{2} c^{2} d^{4} + 9 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{5} c^{4} d^{5} - 2 \, a b^{4} c^{3} d^{6} + a^{2} b^{3} c^{2} d^{7} + {\left (b^{5} c^{2} d^{7} - 2 \, a b^{4} c d^{8} + a^{2} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (b^{5} c^{3} d^{6} - 2 \, a b^{4} c^{2} d^{7} + a^{2} b^{3} c d^{8}\right )} x\right )}}, -\frac {3 \, {\left (35 \, b^{4} c^{6} - 60 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} + 4 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4} + {\left (35 \, b^{4} c^{4} d^{2} - 60 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} + 4 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{5} d - 60 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} + 4 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c 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 (105 \, b^{4} c^{5} d - 145 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} + 9 \, a^{3} b c^{2} d^{4} - 6 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x^{3} + 3 \, {\left (7 \, b^{4} c^{3} d^{3} - 11 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + 3 \, a^{3} b d^{6}\right )} x^{2} + 2 \, {\left (70 \, b^{4} c^{4} d^{2} - 99 \, a b^{3} c^{3} d^{3} + 12 \, a^{2} b^{2} c^{2} d^{4} + 9 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{5} c^{4} d^{5} - 2 \, a b^{4} c^{3} d^{6} + a^{2} b^{3} c^{2} d^{7} + {\left (b^{5} c^{2} d^{7} - 2 \, a b^{4} c d^{8} + a^{2} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (b^{5} c^{3} d^{6} - 2 \, a b^{4} c^{2} d^{7} + a^{2} b^{3} c d^{8}\right )} x\right )}}\right ] \]
[1/48*(3*(35*b^4*c^6 - 60*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 + 4*a^3*b*c^3*d ^3 + 3*a^4*c^2*d^4 + (35*b^4*c^4*d^2 - 60*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d ^4 + 4*a^3*b*c*d^5 + 3*a^4*d^6)*x^2 + 2*(35*b^4*c^5*d - 60*a*b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 + 4*a^3*b*c^2*d^4 + 3*a^4*c*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*(105*b^4*c^ 5*d - 145*a*b^3*c^4*d^2 + 15*a^2*b^2*c^3*d^3 + 9*a^3*b*c^2*d^4 - 6*(b^4*c^ 2*d^4 - 2*a*b^3*c*d^5 + a^2*b^2*d^6)*x^3 + 3*(7*b^4*c^3*d^3 - 11*a*b^3*c^2 *d^4 + a^2*b^2*c*d^5 + 3*a^3*b*d^6)*x^2 + 2*(70*b^4*c^4*d^2 - 99*a*b^3*c^3 *d^3 + 12*a^2*b^2*c^2*d^4 + 9*a^3*b*c*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c)) /(b^5*c^4*d^5 - 2*a*b^4*c^3*d^6 + a^2*b^3*c^2*d^7 + (b^5*c^2*d^7 - 2*a*b^4 *c*d^8 + a^2*b^3*d^9)*x^2 + 2*(b^5*c^3*d^6 - 2*a*b^4*c^2*d^7 + a^2*b^3*c*d ^8)*x), -1/24*(3*(35*b^4*c^6 - 60*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 + 4*a^3 *b*c^3*d^3 + 3*a^4*c^2*d^4 + (35*b^4*c^4*d^2 - 60*a*b^3*c^3*d^3 + 18*a^2*b ^2*c^2*d^4 + 4*a^3*b*c*d^5 + 3*a^4*d^6)*x^2 + 2*(35*b^4*c^5*d - 60*a*b^3*c ^4*d^2 + 18*a^2*b^2*c^3*d^3 + 4*a^3*b*c^2*d^4 + 3*a^4*c*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*(105*b^4*c^5*d - 145*a *b^3*c^4*d^2 + 15*a^2*b^2*c^3*d^3 + 9*a^3*b*c^2*d^4 - 6*(b^4*c^2*d^4 - 2*a *b^3*c*d^5 + a^2*b^2*d^6)*x^3 + 3*(7*b^4*c^3*d^3 - 11*a*b^3*c^2*d^4 + a...
\[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{4}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 508 vs. \(2 (226) = 452\).
Time = 0.39 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.97 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{7} c^{2} d^{6} - 2 \, a b^{6} c d^{7} + a^{2} b^{5} d^{8}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}} - \frac {7 \, b^{8} c^{3} d^{5} - 5 \, a b^{7} c^{2} d^{6} - 11 \, a^{2} b^{6} c d^{7} + 9 \, a^{3} b^{5} d^{8}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} - \frac {4 \, {\left (35 \, b^{9} c^{4} d^{4} - 60 \, a b^{8} c^{3} d^{5} + 18 \, a^{2} b^{7} c^{2} d^{6} + 12 \, a^{3} b^{6} c d^{7} - 9 \, a^{4} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{10} c^{5} d^{3} - 95 \, a b^{9} c^{4} d^{4} + 78 \, a^{2} b^{8} c^{3} d^{5} - 14 \, a^{3} b^{7} c^{2} d^{6} - 9 \, a^{4} b^{6} c d^{7} + 5 \, a^{5} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} b d^{4} {\left | b \right |}} \]
1/12*((3*(b*x + a)*(2*(b^7*c^2*d^6 - 2*a*b^6*c*d^7 + a^2*b^5*d^8)*(b*x + a )/(b^7*c^2*d^7*abs(b) - 2*a*b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)) - (7*b^ 8*c^3*d^5 - 5*a*b^7*c^2*d^6 - 11*a^2*b^6*c*d^7 + 9*a^3*b^5*d^8)/(b^7*c^2*d ^7*abs(b) - 2*a*b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b))) - 4*(35*b^9*c^4*d^ 4 - 60*a*b^8*c^3*d^5 + 18*a^2*b^7*c^2*d^6 + 12*a^3*b^6*c*d^7 - 9*a^4*b^5*d ^8)/(b^7*c^2*d^7*abs(b) - 2*a*b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)))*(b*x + a) - 3*(35*b^10*c^5*d^3 - 95*a*b^9*c^4*d^4 + 78*a^2*b^8*c^3*d^5 - 14*a^ 3*b^7*c^2*d^6 - 9*a^4*b^6*c*d^7 + 5*a^5*b^5*d^8)/(b^7*c^2*d^7*abs(b) - 2*a *b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)))*sqrt(b*x + a)/(b^2*c + (b*x + a)* b*d - a*b*d)^(3/2) - 1/4*(35*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*log(abs(-sq rt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b *d^4*abs(b))
Timed out. \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^4}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]