Integrand size = 22, antiderivative size = 250 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 a x^3}{b (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}-\frac {2 c (b c+a d) x^2 \sqrt {a+b x}}{b d (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right )-2 b d \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) x\right )}{4 b^3 d^3 (b c-a d)^2}+\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{7/2} d^{7/2}} \]
3/4*(5*a^2*d^2+6*a*b*c*d+5*b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/ (d*x+c)^(1/2))/b^(7/2)/d^(7/2)+2*a*x^3/b/(-a*d+b*c)/(b*x+a)^(1/2)/(d*x+c)^ (1/2)-2*c*(a*d+b*c)*x^2*(b*x+a)^(1/2)/b/d/(-a*d+b*c)^2/(d*x+c)^(1/2)-1/4*( (a*d+b*c)*(15*a^2*d^2-22*a*b*c*d+15*b^2*c^2)-2*b*d*(5*a^2*d^2-2*a*b*c*d+5* b^2*c^2)*x)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^3/d^3/(-a*d+b*c)^2
Time = 0.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {-15 a^4 d^3 (c+d x)+a^3 b d^2 \left (7 c^2+2 c d x-5 d^2 x^2\right )+b^4 c^2 x \left (-15 c^2-5 c d x+2 d^2 x^2\right )+a b^3 c \left (-15 c^3+2 c^2 d x+5 c d^2 x^2-4 d^3 x^3\right )+a^2 b^2 d \left (7 c^3+10 c^2 d x+5 c d^2 x^2+2 d^3 x^3\right )}{4 b^3 d^3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {3 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{4 b^{7/2} d^{7/2}} \]
(-15*a^4*d^3*(c + d*x) + a^3*b*d^2*(7*c^2 + 2*c*d*x - 5*d^2*x^2) + b^4*c^2 *x*(-15*c^2 - 5*c*d*x + 2*d^2*x^2) + a*b^3*c*(-15*c^3 + 2*c^2*d*x + 5*c*d^ 2*x^2 - 4*d^3*x^3) + a^2*b^2*d*(7*c^3 + 10*c^2*d*x + 5*c*d^2*x^2 + 2*d^3*x ^3))/(4*b^3*d^3*(b*c - a*d)^2*Sqrt[a + b*x]*Sqrt[c + d*x]) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(4*b^(7/2)*d^(7/2))
Time = 0.40 (sec) , antiderivative size = 279, 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.
| \(\displaystyle \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {2 \int \frac {x^2 (6 a c-(b c-5 a d) x)}{2 \sqrt {a+b x} (c+d x)^{3/2}}dx}{b (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\int \frac {x^2 (6 a c-(b c-5 a d) x)}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{b (b c-a d)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{d \sqrt {c+d x} (b c-a d)}-\frac {2 \int \frac {x \left (4 a c (b c+a d)+\left (5 b^2 c^2-2 a b d c+5 a^2 d^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{d \sqrt {c+d x} (b c-a d)}-\frac {\int \frac {x \left (4 a c (b c+a d)+\left (5 b^2 c^2-2 a b d c+5 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{d (b c-a d)}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{d \sqrt {c+d x} (b c-a d)}-\frac {\frac {3 (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{8 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^2 d^2}}{d (b c-a d)}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{d \sqrt {c+d x} (b c-a d)}-\frac {\frac {3 (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{4 b^2 d^2}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^2 d^2}}{d (b c-a d)}}{b (b c-a d)}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 a x^3}{b \sqrt {a+b x} \sqrt {c+d x} (b c-a d)}-\frac {\frac {2 c x^2 \sqrt {a+b x} (a d+b c)}{d \sqrt {c+d x} (b c-a d)}-\frac {\frac {3 (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{5/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^2 d^2}}{d (b c-a d)}}{b (b c-a d)}\) |
(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - ((2*c*(b*c + a*d)* x^2*Sqrt[a + b*x])/(d*(b*c - a*d)*Sqrt[c + d*x]) - (-1/4*(Sqrt[a + b*x]*Sq rt[c + d*x]*((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2) - 2*b*d*(5 *b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*x))/(b^2*d^2) + (3*(b*c - a*d)^2*(5*b^2* c^2 + 6*a*b*c*d + 5*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)))/(b*(b*c - a*d))
3.8.76.3.1 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. \(1368\) vs. \(2(222)=444\).
Time = 0.59 (sec) , antiderivative size = 1369, normalized size of antiderivative = 5.48
1/8*(-30*a^4*d^4*x*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*b^4*c^4*x*((b*x+ a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a^4*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^( 1/2)-30*a*b^3*c^4*((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^4*b*d^5*x^2+1 5*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^5*c^4*d*x^2-12*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^4*b*c^2*d^3-6*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^2*c^3*d^2-12*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^2*b^3*c^4* d-10*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*d^4*x^2-10*(b*d)^(1/2)*((b* x+a)*(d*x+c))^(1/2)*b^4*c^3*d*x^2+14*(b*d)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a ^3*b*c^2*d^2+4*a^2*b^2*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4*b^4*c ^2*d^2*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-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^2*c*d^4*x^2-6*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^3*c^2*d^3*x^2-12*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*b^4*c^3*d^2*x^2+3*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*c*d^4*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^3*b^2* c^2*d^3*x-18*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. 593 vs. \(2 (223) = 446\).
Time = 0.53 (sec) , antiderivative size = 1200, normalized size of antiderivative = 4.80 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\left [\frac {3 \, {\left (5 \, a b^{4} c^{5} - 4 \, a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} - 4 \, a^{4} b c^{2} d^{3} + 5 \, a^{5} c d^{4} + {\left (5 \, b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x^{2} + {\left (5 \, b^{5} c^{5} + a b^{4} c^{4} d - 6 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + 5 \, a^{5} 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 (15 \, a b^{4} c^{4} d - 7 \, a^{2} b^{3} c^{3} d^{2} - 7 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 2 \, {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{3} + 5 \, {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (15 \, b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} - 10 \, a^{2} b^{3} c^{2} d^{3} - 2 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, {\left (a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x\right )}}, -\frac {3 \, {\left (5 \, a b^{4} c^{5} - 4 \, a^{2} b^{3} c^{4} d - 2 \, a^{3} b^{2} c^{3} d^{2} - 4 \, a^{4} b c^{2} d^{3} + 5 \, a^{5} c d^{4} + {\left (5 \, b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} - 2 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x^{2} + {\left (5 \, b^{5} c^{5} + a b^{4} c^{4} d - 6 \, a^{2} b^{3} c^{3} d^{2} - 6 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} + 5 \, a^{5} 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 (15 \, a b^{4} c^{4} d - 7 \, a^{2} b^{3} c^{3} d^{2} - 7 \, a^{3} b^{2} c^{2} d^{3} + 15 \, a^{4} b c d^{4} - 2 \, {\left (b^{5} c^{2} d^{3} - 2 \, a b^{4} c d^{4} + a^{2} b^{3} d^{5}\right )} x^{3} + 5 \, {\left (b^{5} c^{3} d^{2} - a b^{4} c^{2} d^{3} - a^{2} b^{3} c d^{4} + a^{3} b^{2} d^{5}\right )} x^{2} + {\left (15 \, b^{5} c^{4} d - 2 \, a b^{4} c^{3} d^{2} - 10 \, a^{2} b^{3} c^{2} d^{3} - 2 \, a^{3} b^{2} c d^{4} + 15 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, {\left (a b^{6} c^{3} d^{4} - 2 \, a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + {\left (b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{2} + {\left (b^{7} c^{3} d^{4} - a b^{6} c^{2} d^{5} - a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x\right )}}\right ] \]
[1/16*(3*(5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2* d^3 + 5*a^5*c*d^4 + (5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4 *a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^2 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c ^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*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*(15*a*b^4* c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 2*(b^5*c^ 2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^3 + 5*(b^5*c^3*d^2 - a*b^4*c^2*d^3 - a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (15*b^5*c^4*d - 2*a*b^4*c^3*d^2 - 10* a^2*b^3*c^2*d^3 - 2*a^3*b^2*c*d^4 + 15*a^4*b*d^5)*x)*sqrt(b*x + a)*sqrt(d* x + c))/(a*b^6*c^3*d^4 - 2*a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + (b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)*x^2 + (b^7*c^3*d^4 - a*b^6*c^2*d^5 - a^2*b^ 5*c*d^6 + a^3*b^4*d^7)*x), -1/8*(3*(5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3* b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4 + (5*b^5*c^4*d - 4*a*b^4*c^3*d ^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^2 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5 *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*(15 *a*b^4*c^4*d - 7*a^2*b^3*c^3*d^2 - 7*a^3*b^2*c^2*d^3 + 15*a^4*b*c*d^4 - 2* (b^5*c^2*d^3 - 2*a*b^4*c*d^4 + a^2*b^3*d^5)*x^3 + 5*(b^5*c^3*d^2 - a*b^...
\[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {x^{4}}{\left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (223) = 446\).
Time = 0.44 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.91 \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=-\frac {4 \, a^{4} d}{{\left (\sqrt {b d} b^{2} c {\left | b \right |} - \sqrt {b d} a b d {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {{\left ({\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{10} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{9} c d^{5} {\left | b \right |} + a^{2} b^{8} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}} - \frac {5 \, b^{11} c^{3} d^{3} {\left | b \right |} + a b^{10} c^{2} d^{4} {\left | b \right |} - 17 \, a^{2} b^{9} c d^{5} {\left | b \right |} + 11 \, a^{3} b^{8} d^{6} {\left | b \right |}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}}\right )} - \frac {15 \, b^{12} c^{4} d^{2} {\left | b \right |} - 12 \, a b^{11} c^{3} d^{3} {\left | b \right |} - 6 \, a^{2} b^{10} c^{2} d^{4} {\left | b \right |} + 20 \, a^{3} b^{9} c d^{5} {\left | b \right |} - 9 \, a^{4} b^{8} d^{6} {\left | b \right |}}{b^{14} c^{2} d^{5} - 2 \, a b^{13} c d^{6} + a^{2} b^{12} d^{7}}\right )} \sqrt {b x + a}}{4 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} - \frac {3 \, {\left (5 \, b^{2} c^{2} + 6 \, a b c d + 5 \, a^{2} d^{2}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{8 \, \sqrt {b d} b^{2} d^{3} {\left | b \right |}} \]
-4*a^4*d/((sqrt(b*d)*b^2*c*abs(b) - sqrt(b*d)*a*b*d*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)) + 1 /4*((b*x + a)*(2*(b^10*c^2*d^4*abs(b) - 2*a*b^9*c*d^5*abs(b) + a^2*b^8*d^6 *abs(b))*(b*x + a)/(b^14*c^2*d^5 - 2*a*b^13*c*d^6 + a^2*b^12*d^7) - (5*b^1 1*c^3*d^3*abs(b) + a*b^10*c^2*d^4*abs(b) - 17*a^2*b^9*c*d^5*abs(b) + 11*a^ 3*b^8*d^6*abs(b))/(b^14*c^2*d^5 - 2*a*b^13*c*d^6 + a^2*b^12*d^7)) - (15*b^ 12*c^4*d^2*abs(b) - 12*a*b^11*c^3*d^3*abs(b) - 6*a^2*b^10*c^2*d^4*abs(b) + 20*a^3*b^9*c*d^5*abs(b) - 9*a^4*b^8*d^6*abs(b))/(b^14*c^2*d^5 - 2*a*b^13* c*d^6 + a^2*b^12*d^7))*sqrt(b*x + a)/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) - 3/8*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*log((sqrt(b*d)*sqrt(b*x + a) - sq rt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(b*d)*b^2*d^3*abs(b))
Timed out. \[ \int \frac {x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\int \frac {x^4}{{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{3/2}} \,d x \]