Integrand size = 22, antiderivative size = 437 \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=-\frac {(b c-a d)^4 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{1024 b^5 d^4}+\frac {(b c-a d)^3 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{1536 b^5 d^3}+\frac {(b c-a d)^2 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{384 b^5 d^2}+\frac {(b c-a d) \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{3/2}}{192 b^4 d^2}+\frac {\left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) (a+b x)^{5/2} (c+d x)^{5/2}}{120 b^3 d^2}-\frac {(7 b c+9 a d) (a+b x)^{5/2} (c+d x)^{7/2}}{84 b^2 d^2}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}+\frac {(b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{1024 b^{11/2} d^{9/2}} \]
1/192*(-a*d+b*c)*(9*a^2*d^2+10*a*b*c*d+5*b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^(3 /2)/b^4/d^2+1/120*(9*a^2*d^2+10*a*b*c*d+5*b^2*c^2)*(b*x+a)^(5/2)*(d*x+c)^( 5/2)/b^3/d^2-1/84*(9*a*d+7*b*c)*(b*x+a)^(5/2)*(d*x+c)^(7/2)/b^2/d^2+1/7*x* (b*x+a)^(5/2)*(d*x+c)^(7/2)/b/d+1/1024*(-a*d+b*c)^5*(9*a^2*d^2+10*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^(11/2)/d ^(9/2)+1/1536*(-a*d+b*c)^3*(9*a^2*d^2+10*a*b*c*d+5*b^2*c^2)*(b*x+a)^(3/2)* (d*x+c)^(1/2)/b^5/d^3+1/384*(-a*d+b*c)^2*(9*a^2*d^2+10*a*b*c*d+5*b^2*c^2)* (b*x+a)^(5/2)*(d*x+c)^(1/2)/b^5/d^2-1/1024*(-a*d+b*c)^4*(9*a^2*d^2+10*a*b* c*d+5*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^5/d^4
Time = 1.02 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.87 \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (945 a^6 d^6-210 a^5 b d^5 (16 c+3 d x)+7 a^4 b^2 d^4 \left (527 c^2+314 c d x+72 d^2 x^2\right )-4 a^3 b^3 d^3 \left (150 c^3+583 c^2 d x+436 c d^2 x^2+108 d^3 x^3\right )+3 a^2 b^4 d^2 \left (-175 c^4+100 c^3 d x+608 c^2 d^2 x^2+496 c d^3 x^3+128 d^4 x^4\right )+10 a b^5 d \left (140 c^5-91 c^4 d x+72 c^3 d^2 x^2+3352 c^2 d^3 x^3+4864 c d^4 x^4+1920 d^5 x^5\right )-5 b^6 \left (105 c^6-70 c^5 d x+56 c^4 d^2 x^2-48 c^3 d^3 x^3-4736 c^2 d^4 x^4-7424 c d^5 x^5-3072 d^6 x^6\right )\right )}{107520 b^5 d^4}+\frac {(b c-a d)^5 \left (5 b^2 c^2+10 a b c d+9 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{1024 b^{11/2} d^{9/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(945*a^6*d^6 - 210*a^5*b*d^5*(16*c + 3*d*x) + 7*a^4*b^2*d^4*(527*c^2 + 314*c*d*x + 72*d^2*x^2) - 4*a^3*b^3*d^3*(150*c^3 + 583*c^2*d*x + 436*c*d^2*x^2 + 108*d^3*x^3) + 3*a^2*b^4*d^2*(-175*c^4 + 100*c^3*d*x + 608*c^2*d^2*x^2 + 496*c*d^3*x^3 + 128*d^4*x^4) + 10*a*b^5*d* (140*c^5 - 91*c^4*d*x + 72*c^3*d^2*x^2 + 3352*c^2*d^3*x^3 + 4864*c*d^4*x^4 + 1920*d^5*x^5) - 5*b^6*(105*c^6 - 70*c^5*d*x + 56*c^4*d^2*x^2 - 48*c^3*d ^3*x^3 - 4736*c^2*d^4*x^4 - 7424*c*d^5*x^5 - 3072*d^6*x^6)))/(107520*b^5*d ^4) + ((b*c - a*d)^5*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*ArcTanh[(Sqrt[b] *Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/(1024*b^(11/2)*d^(9/2))
Time = 0.37 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.80, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {101, 27, 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.
| \(\displaystyle \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 101 |
| \(\displaystyle \frac {\int -\frac {1}{2} (a+b x)^{3/2} (c+d x)^{5/2} (2 a c+(7 b c+9 a d) x)dx}{7 b d}+\frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\int (a+b x)^{3/2} (c+d x)^{5/2} (2 a c+(7 b c+9 a d) x)dx}{14 b d}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \int (a+b x)^{3/2} (c+d x)^{5/2}dx}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \int (a+b x)^{3/2} (c+d x)^{3/2}dx}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \int (a+b x)^{3/2} \sqrt {c+d x}dx}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}}dx}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}}dx}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{d}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x (a+b x)^{5/2} (c+d x)^{7/2}}{7 b d}-\frac {\frac {(a+b x)^{5/2} (c+d x)^{7/2} (9 a d+7 b c)}{6 b d}-\frac {7 \left (9 a^2 d^2+10 a b c d+5 b^2 c^2\right ) \left (\frac {(b c-a d) \left (\frac {3 (b c-a d) \left (\frac {(b c-a d) \left (\frac {(a+b x)^{3/2} \sqrt {c+d x}}{2 d}-\frac {3 (b c-a d) \left (\frac {\sqrt {a+b x} \sqrt {c+d x}}{d}-\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} d^{3/2}}\right )}{4 d}\right )}{6 b}+\frac {(a+b x)^{5/2} \sqrt {c+d x}}{3 b}\right )}{8 b}+\frac {(a+b x)^{5/2} (c+d x)^{3/2}}{4 b}\right )}{2 b}+\frac {(a+b x)^{5/2} (c+d x)^{5/2}}{5 b}\right )}{12 b d}}{14 b d}\) |
(x*(a + b*x)^(5/2)*(c + d*x)^(7/2))/(7*b*d) - (((7*b*c + 9*a*d)*(a + b*x)^ (5/2)*(c + d*x)^(7/2))/(6*b*d) - (7*(5*b^2*c^2 + 10*a*b*c*d + 9*a^2*d^2)*( ((a + b*x)^(5/2)*(c + d*x)^(5/2))/(5*b) + ((b*c - a*d)*(((a + b*x)^(5/2)*( c + d*x)^(3/2))/(4*b) + (3*(b*c - a*d)*(((a + b*x)^(5/2)*Sqrt[c + d*x])/(3 *b) + ((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*b)))/(8*b) ))/(2*b)))/(12*b*d))/(14*b*d)
3.7.16.3.1 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_))*((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*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 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. \(1320\) vs. \(2(381)=762\).
Time = 0.55 (sec) , antiderivative size = 1321, normalized size of antiderivative = 3.02
-1/215040*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(-97280*a*b^5*c*d^5*x^4*((b*x+a)*(d* x+c))^(1/2)*(b*d)^(1/2)-2976*a^2*b^4*c*d^5*x^3*((b*x+a)*(d*x+c))^(1/2)*(b* d)^(1/2)-67040*a*b^5*c^2*d^4*x^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+3488* a^3*b^3*c*d^5*x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-3648*a^2*b^4*c^2*d^4 *x^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-1440*a*b^5*c^3*d^3*x^2*((b*x+a)*( d*x+c))^(1/2)*(b*d)^(1/2)+6720*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b*c *d^5-7378*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c^2*d^4+1200*((b*x+a )*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^3*d^3+1050*((b*x+a)*(d*x+c))^(1/2)* (b*d)^(1/2)*a^2*b^4*c^4*d^2+1260*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^5*b *d^6*x-700*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*b^6*c^5*d*x-1890*((b*x+a)*( d*x+c))^(1/2)*(b*d)^(1/2)*a^6*d^6+1050*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2) *b^6*c^6-1575*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^3*c^3*d^4-525*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^4*c^4*d^3-525*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^7*c^7+945*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^7*d^7-30720*b^6*d^6*x^6*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4725*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 ^2*c^2*d^5-4396*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^4*b^2*c*d^5*x+4664*( (b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b^3*c^2*d^4*x-945*ln(1/2*(2*b*d*...
Time = 0.29 (sec) , antiderivative size = 1110, normalized size of antiderivative = 2.54 \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\left [-\frac {105 \, {\left (5 \, b^{7} c^{7} - 15 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 15 \, a^{4} b^{3} c^{3} d^{4} - 45 \, a^{5} b^{2} c^{2} d^{5} + 35 \, a^{6} b c d^{6} - 9 \, a^{7} d^{7}\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 (15360 \, b^{7} d^{7} x^{6} - 525 \, b^{7} c^{6} d + 1400 \, a b^{6} c^{5} d^{2} - 525 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} + 3689 \, a^{4} b^{3} c^{2} d^{5} - 3360 \, a^{5} b^{2} c d^{6} + 945 \, a^{6} b d^{7} + 1280 \, {\left (29 \, b^{7} c d^{6} + 15 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (185 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (15 \, b^{7} c^{3} d^{4} + 2095 \, a b^{6} c^{2} d^{5} + 93 \, a^{2} b^{5} c d^{6} - 27 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (35 \, b^{7} c^{4} d^{3} - 90 \, a b^{6} c^{3} d^{4} - 228 \, a^{2} b^{5} c^{2} d^{5} + 218 \, a^{3} b^{4} c d^{6} - 63 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (175 \, b^{7} c^{5} d^{2} - 455 \, a b^{6} c^{4} d^{3} + 150 \, a^{2} b^{5} c^{3} d^{4} - 1166 \, a^{3} b^{4} c^{2} d^{5} + 1099 \, a^{4} b^{3} c d^{6} - 315 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{430080 \, b^{6} d^{5}}, -\frac {105 \, {\left (5 \, b^{7} c^{7} - 15 \, a b^{6} c^{6} d + 9 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 15 \, a^{4} b^{3} c^{3} d^{4} - 45 \, a^{5} b^{2} c^{2} d^{5} + 35 \, a^{6} b c d^{6} - 9 \, a^{7} d^{7}\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 (15360 \, b^{7} d^{7} x^{6} - 525 \, b^{7} c^{6} d + 1400 \, a b^{6} c^{5} d^{2} - 525 \, a^{2} b^{5} c^{4} d^{3} - 600 \, a^{3} b^{4} c^{3} d^{4} + 3689 \, a^{4} b^{3} c^{2} d^{5} - 3360 \, a^{5} b^{2} c d^{6} + 945 \, a^{6} b d^{7} + 1280 \, {\left (29 \, b^{7} c d^{6} + 15 \, a b^{6} d^{7}\right )} x^{5} + 128 \, {\left (185 \, b^{7} c^{2} d^{5} + 380 \, a b^{6} c d^{6} + 3 \, a^{2} b^{5} d^{7}\right )} x^{4} + 16 \, {\left (15 \, b^{7} c^{3} d^{4} + 2095 \, a b^{6} c^{2} d^{5} + 93 \, a^{2} b^{5} c d^{6} - 27 \, a^{3} b^{4} d^{7}\right )} x^{3} - 8 \, {\left (35 \, b^{7} c^{4} d^{3} - 90 \, a b^{6} c^{3} d^{4} - 228 \, a^{2} b^{5} c^{2} d^{5} + 218 \, a^{3} b^{4} c d^{6} - 63 \, a^{4} b^{3} d^{7}\right )} x^{2} + 2 \, {\left (175 \, b^{7} c^{5} d^{2} - 455 \, a b^{6} c^{4} d^{3} + 150 \, a^{2} b^{5} c^{3} d^{4} - 1166 \, a^{3} b^{4} c^{2} d^{5} + 1099 \, a^{4} b^{3} c d^{6} - 315 \, a^{5} b^{2} d^{7}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{215040 \, b^{6} d^{5}}\right ] \]
[-1/430080*(105*(5*b^7*c^7 - 15*a*b^6*c^6*d + 9*a^2*b^5*c^5*d^2 + 5*a^3*b^ 4*c^4*d^3 + 15*a^4*b^3*c^3*d^4 - 45*a^5*b^2*c^2*d^5 + 35*a^6*b*c*d^6 - 9*a ^7*d^7)*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*(15360*b^7*d^7*x^6 - 525*b^7*c^6*d + 1400*a*b^6*c^5*d^2 - 5 25*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 + 3689*a^4*b^3*c^2*d^5 - 3360*a^5 *b^2*c*d^6 + 945*a^6*b*d^7 + 1280*(29*b^7*c*d^6 + 15*a*b^6*d^7)*x^5 + 128* (185*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 3*a^2*b^5*d^7)*x^4 + 16*(15*b^7*c^3*d ^4 + 2095*a*b^6*c^2*d^5 + 93*a^2*b^5*c*d^6 - 27*a^3*b^4*d^7)*x^3 - 8*(35*b ^7*c^4*d^3 - 90*a*b^6*c^3*d^4 - 228*a^2*b^5*c^2*d^5 + 218*a^3*b^4*c*d^6 - 63*a^4*b^3*d^7)*x^2 + 2*(175*b^7*c^5*d^2 - 455*a*b^6*c^4*d^3 + 150*a^2*b^5 *c^3*d^4 - 1166*a^3*b^4*c^2*d^5 + 1099*a^4*b^3*c*d^6 - 315*a^5*b^2*d^7)*x) *sqrt(b*x + a)*sqrt(d*x + c))/(b^6*d^5), -1/215040*(105*(5*b^7*c^7 - 15*a* b^6*c^6*d + 9*a^2*b^5*c^5*d^2 + 5*a^3*b^4*c^4*d^3 + 15*a^4*b^3*c^3*d^4 - 4 5*a^5*b^2*c^2*d^5 + 35*a^6*b*c*d^6 - 9*a^7*d^7)*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*(15360*b^7*d^7*x^6 - 525*b^7*c^6*d + 1 400*a*b^6*c^5*d^2 - 525*a^2*b^5*c^4*d^3 - 600*a^3*b^4*c^3*d^4 + 3689*a^4*b ^3*c^2*d^5 - 3360*a^5*b^2*c*d^6 + 945*a^6*b*d^7 + 1280*(29*b^7*c*d^6 + 15* a*b^6*d^7)*x^5 + 128*(185*b^7*c^2*d^5 + 380*a*b^6*c*d^6 + 3*a^2*b^5*d^7...
\[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\int x^{2} \left (a + b x\right )^{\frac {3}{2}} \left (c + d x\right )^{\frac {5}{2}}\, dx \]
Exception generated. \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\text {Exception raised: ValueError} \]
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. 3513 vs. \(2 (381) = 762\).
Time = 0.70 (sec) , antiderivative size = 3513, normalized size of antiderivative = 8.04 \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\text {Too large to display} \]
1/107520*(56*(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^23*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))*c^2*abs(b) + 4480*(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)*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*d^2))*a^2*c^2*abs(b)/b^2 + 1120*(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))*sqrt(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...
Timed out. \[ \int x^2 (a+b x)^{3/2} (c+d x)^{5/2} \, dx=\int x^2\,{\left (a+b\,x\right )}^{3/2}\,{\left (c+d\,x\right )}^{5/2} \,d x \]