Integrand size = 20, antiderivative size = 315 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=-\frac {(b c-a d)^4 (7 b c+5 a d) \sqrt {a+b x} \sqrt {c+d x}}{512 b^3 d^4}+\frac {(b c-a d)^3 (7 b c+5 a d) (a+b x)^{3/2} \sqrt {c+d x}}{768 b^3 d^3}-\frac {(b c-a d)^2 (7 b c+5 a d) (a+b x)^{5/2} \sqrt {c+d x}}{960 b^3 d^2}-\frac {(b c-a d) (7 b c+5 a d) (a+b x)^{7/2} \sqrt {c+d x}}{160 b^3 d}-\frac {(7 b c+5 a d) (a+b x)^{7/2} (c+d x)^{3/2}}{60 b^2 d}+\frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \]
-1/60*(5*a*d+7*b*c)*(b*x+a)^(7/2)*(d*x+c)^(3/2)/b^2/d+1/6*(b*x+a)^(7/2)*(d *x+c)^(5/2)/b/d+1/512*(-a*d+b*c)^5*(5*a*d+7*b*c)*arctanh(d^(1/2)*(b*x+a)^( 1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(7/2)/d^(9/2)+1/768*(-a*d+b*c)^3*(5*a*d+7*b* c)*(b*x+a)^(3/2)*(d*x+c)^(1/2)/b^3/d^3-1/960*(-a*d+b*c)^2*(5*a*d+7*b*c)*(b *x+a)^(5/2)*(d*x+c)^(1/2)/b^3/d^2-1/160*(-a*d+b*c)*(5*a*d+7*b*c)*(b*x+a)^( 7/2)*(d*x+c)^(1/2)/b^3/d-1/512*(-a*d+b*c)^4*(5*a*d+7*b*c)*(b*x+a)^(1/2)*(d *x+c)^(1/2)/b^3/d^4
Time = 0.65 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.93 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (75 a^5 d^5-5 a^4 b d^4 (49 c+10 d x)+10 a^3 b^2 d^3 \left (15 c^2+16 c d x+4 d^2 x^2\right )+6 a^2 b^3 d^2 \left (-91 c^3+58 c^2 d x+564 c d^2 x^2+360 d^3 x^3\right )+a b^4 d \left (415 c^4-272 c^3 d x+216 c^2 d^2 x^2+4448 c d^3 x^3+3200 d^4 x^4\right )+b^5 \left (-105 c^5+70 c^4 d x-56 c^3 d^2 x^2+48 c^2 d^3 x^3+1664 c d^4 x^4+1280 d^5 x^5\right )\right )}{7680 b^3 d^4}+\frac {(b c-a d)^5 (7 b c+5 a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{512 b^{7/2} d^{9/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(75*a^5*d^5 - 5*a^4*b*d^4*(49*c + 10*d*x) + 1 0*a^3*b^2*d^3*(15*c^2 + 16*c*d*x + 4*d^2*x^2) + 6*a^2*b^3*d^2*(-91*c^3 + 5 8*c^2*d*x + 564*c*d^2*x^2 + 360*d^3*x^3) + a*b^4*d*(415*c^4 - 272*c^3*d*x + 216*c^2*d^2*x^2 + 4448*c*d^3*x^3 + 3200*d^4*x^4) + b^5*(-105*c^5 + 70*c^ 4*d*x - 56*c^3*d^2*x^2 + 48*c^2*d^3*x^3 + 1664*c*d^4*x^4 + 1280*d^5*x^5))) /(7680*b^3*d^4) + ((b*c - a*d)^5*(7*b*c + 5*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(512*b^(7/2)*d^(9/2))
Time = 0.30 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {90, 60, 60, 60, 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.
| \(\displaystyle \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \int (a+b x)^{5/2} (c+d x)^{3/2}dx}{12 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \int (a+b x)^{5/2} \sqrt {c+d x}dx}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}}dx}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(a+b x)^{7/2} (c+d x)^{5/2}}{6 b d}-\frac {(5 a d+7 b c) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 d}-\frac {5 (b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 d}\right )}{8 b}+\frac {(a+b x)^{7/2} \sqrt {c+d x}}{4 b}\right )}{10 b}+\frac {(a+b x)^{7/2} (c+d x)^{3/2}}{5 b}\right )}{12 b d}\) |
((a + b*x)^(7/2)*(c + d*x)^(5/2))/(6*b*d) - ((7*b*c + 5*a*d)*(((a + b*x)^( 7/2)*(c + d*x)^(3/2))/(5*b) + (3*(b*c - a*d)*(((a + b*x)^(7/2)*Sqrt[c + d* x])/(4*b) + ((b*c - a*d)*(((a + b*x)^(5/2)*Sqrt[c + d*x])/(3*d) - (5*(b*c - a*d)*(((a + b*x)^(3/2)*Sqrt[c + d*x])/(2*d) - (3*(b*c - a*d)*((Sqrt[a + b*x]*Sqrt[c + d*x])/d - ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt [b]*Sqrt[c + d*x])])/(Sqrt[b]*d^(3/2))))/(4*d)))/(6*d)))/(8*b)))/(10*b)))/ (12*b*d)
3.7.57.3.1 Defintions of rubi rules used
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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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. \(1036\) vs. \(2(265)=530\).
Time = 1.59 (sec) , antiderivative size = 1037, normalized size of antiderivative = 3.29
-1/15360*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(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^6*d^6-105*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^6*c^6-150*((b*x +a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*d^5+210*((b*x+a)*(d*x+c))^(1/2)*(b*d)^( 1/2)*b^5*c^5+225*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^4+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^3*b^3*c^3*d^3-675*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 *d^2+450*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*d+100*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b*d^5*x +490*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b*c*d^4-300*((b*x+a)*(d*x+c)) ^(1/2)*(b*d)^(1/2)*a^3*b^2*c^2*d^3-830*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2) *a*b^4*c^4*d+1092*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^2*b^3*c^3*d^2-2560 *b^5*d^5*x^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-270*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^5-8896*a* b^4*c*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6768*a^2*b^3*c*d^4*x^2*( (b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-432*a*b^4*c^2*d^3*x^2*((b*x+a)*(d*x+c)) ^(1/2)*(b*d)^(1/2)+544*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a*b^4*c^3*d^2*x -6400*a*b^4*d^5*x^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-3328*b^5*c*d^4*x^4 *((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-4320*a^2*b^3*d^5*x^3*((b*x+a)*(d*x...
Time = 0.28 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.84 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\left [-\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} - 105 \, b^{6} c^{5} d + 415 \, a b^{5} c^{4} d^{2} - 546 \, a^{2} b^{4} c^{3} d^{3} + 150 \, a^{3} b^{3} c^{2} d^{4} - 245 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (13 \, b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 278 \, a b^{5} c d^{5} + 135 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 27 \, a b^{5} c^{2} d^{4} - 423 \, a^{2} b^{4} c d^{5} - 5 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 136 \, a b^{5} c^{3} d^{3} + 174 \, a^{2} b^{4} c^{2} d^{4} + 80 \, a^{3} b^{3} c d^{5} - 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{30720 \, b^{4} d^{5}}, -\frac {15 \, {\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\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 (1280 \, b^{6} d^{6} x^{5} - 105 \, b^{6} c^{5} d + 415 \, a b^{5} c^{4} d^{2} - 546 \, a^{2} b^{4} c^{3} d^{3} + 150 \, a^{3} b^{3} c^{2} d^{4} - 245 \, a^{4} b^{2} c d^{5} + 75 \, a^{5} b d^{6} + 128 \, {\left (13 \, b^{6} c d^{5} + 25 \, a b^{5} d^{6}\right )} x^{4} + 16 \, {\left (3 \, b^{6} c^{2} d^{4} + 278 \, a b^{5} c d^{5} + 135 \, a^{2} b^{4} d^{6}\right )} x^{3} - 8 \, {\left (7 \, b^{6} c^{3} d^{3} - 27 \, a b^{5} c^{2} d^{4} - 423 \, a^{2} b^{4} c d^{5} - 5 \, a^{3} b^{3} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{6} c^{4} d^{2} - 136 \, a b^{5} c^{3} d^{3} + 174 \, a^{2} b^{4} c^{2} d^{4} + 80 \, a^{3} b^{3} c d^{5} - 25 \, a^{4} b^{2} d^{6}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{15360 \, b^{4} d^{5}}\right ] \]
[-1/30720*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^3*b^ 3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*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)*s qrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(1280* b^6*d^6*x^5 - 105*b^6*c^5*d + 415*a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 15 0*a^3*b^3*c^2*d^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d^5 + 25*a*b^5*d^6)*x^4 + 16*(3*b^6*c^2*d^4 + 278*a*b^5*c*d^5 + 135*a^2*b^4*d^6 )*x^3 - 8*(7*b^6*c^3*d^3 - 27*a*b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^ 3*d^6)*x^2 + 2*(35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2*d^4 + 80*a^3*b^3*c*d^5 - 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d ^5), -1/15360*(15*(7*b^6*c^6 - 30*a*b^5*c^5*d + 45*a^2*b^4*c^4*d^2 - 20*a^ 3*b^3*c^3*d^3 - 15*a^4*b^2*c^2*d^4 + 18*a^5*b*c*d^5 - 5*a^6*d^6)*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*(1280*b^6*d^6*x^5 - 1 05*b^6*c^5*d + 415*a*b^5*c^4*d^2 - 546*a^2*b^4*c^3*d^3 + 150*a^3*b^3*c^2*d ^4 - 245*a^4*b^2*c*d^5 + 75*a^5*b*d^6 + 128*(13*b^6*c*d^5 + 25*a*b^5*d^6)* x^4 + 16*(3*b^6*c^2*d^4 + 278*a*b^5*c*d^5 + 135*a^2*b^4*d^6)*x^3 - 8*(7*b^ 6*c^3*d^3 - 27*a*b^5*c^2*d^4 - 423*a^2*b^4*c*d^5 - 5*a^3*b^3*d^6)*x^2 + 2* (35*b^6*c^4*d^2 - 136*a*b^5*c^3*d^3 + 174*a^2*b^4*c^2*d^4 + 80*a^3*b^3*c*d ^5 - 25*a^4*b^2*d^6)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*d^5)]
\[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}\, dx \]
Exception generated. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Exception raised: ValueError} \]
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. 2375 vs. \(2 (265) = 530\).
Time = 0.61 (sec) , antiderivative size = 2375, normalized size of antiderivative = 7.54 \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\text {Too large to display} \]
1/7680*(120*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a) *(6*(b*x + a)/b^3 + (b^12*c*d^5 - 25*a*b^11*d^6)/(b^14*d^6)) - (5*b^13*c^2 *d^4 + 14*a*b^12*c*d^5 - 163*a^2*b^11*d^6)/(b^14*d^6)) + 3*(5*b^14*c^3*d^3 + 9*a*b^13*c^2*d^4 + 15*a^2*b^12*c*d^5 - 93*a^3*b^11*d^6)/(b^14*d^6))*sqr t(b*x + a) + 3*(5*b^4*c^4 + 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 20*a^3*b*c *d^3 - 35*a^4*d^4)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3))*a*c*abs(b) + 960*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*c*d^ 3 - 13*a*b^5*d^4)/(b^7*d^4)) - 3*(b^7*c^2*d^2 + 2*a*b^6*c*d^3 - 11*a^2*b^5 *d^4)/(b^7*d^4)) - 3*(b^3*c^3 + a*b^2*c^2*d + 3*a^2*b*c*d^2 - 5*a^3*d^3)*l og(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(s qrt(b*d)*b*d^2))*a^2*c*abs(b)/b + 4*(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*( 2*(4*(b*x + a)*(6*(b*x + a)*(8*(b*x + a)/b^4 + (b^20*c*d^7 - 41*a*b^19*d^8 )/(b^23*d^8)) - (7*b^21*c^2*d^6 + 26*a*b^20*c*d^7 - 513*a^2*b^19*d^8)/(b^2 3*d^8)) + 5*(7*b^22*c^3*d^5 + 19*a*b^21*c^2*d^6 + 37*a^2*b^20*c*d^7 - 447* a^3*b^19*d^8)/(b^23*d^8))*(b*x + a) - 15*(7*b^23*c^4*d^4 + 12*a*b^22*c^3*d ^5 + 18*a^2*b^21*c^2*d^6 + 28*a^3*b^20*c*d^7 - 193*a^4*b^19*d^8)/(b^23*d^8 ))*sqrt(b*x + a) - 15*(7*b^5*c^5 + 5*a*b^4*c^4*d + 6*a^2*b^3*c^3*d^2 + 10* a^3*b^2*c^2*d^3 + 35*a^4*b*c*d^4 - 63*a^5*d^5)*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))/(sqrt(b*d)*b^3*d^4))*b*c*...
Timed out. \[ \int x (a+b x)^{5/2} (c+d x)^{3/2} \, dx=\int x\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{3/2} \,d x \]