Integrand size = 22, antiderivative size = 189 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {4 a^2 c}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 c \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 c \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \sqrt {a+b x}}{3 b d (b c-a d)^4 \sqrt {c+d x}} \]
-2/3*x^3/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-4*a^2*c/b^2/(-a*d+b*c)^2/( d*x+c)^(3/2)/(b*x+a)^(1/2)-4/3*c*(3*a^2*d^2+b^2*c^2)*(b*x+a)^(1/2)/b^2/d/( -a*d+b*c)^3/(d*x+c)^(3/2)+4/3*c*(-3*a^2*d^2-6*a*b*c*d+b^2*c^2)*(b*x+a)^(1/ 2)/b/d/(-a*d+b*c)^4/(d*x+c)^(1/2)
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (c^3-\frac {9 a c^2 (c+d x)}{a+b x}-\frac {9 a^2 c (c+d x)^2}{(a+b x)^2}+\frac {a^3 (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \]
(2*(a + b*x)^(3/2)*(c^3 - (9*a*c^2*(c + d*x))/(a + b*x) - (9*a^2*c*(c + d* x)^2)/(a + b*x)^2 + (a^3*(c + d*x)^3)/(a + b*x)^3))/(3*(b*c - a*d)^4*(c + d*x)^(3/2))
Time = 0.31 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.12, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 100, 27, 87, 48}
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.
| \(\displaystyle \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {2 c \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{5/2}}dx}{b c-a d}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {2 c \left (\frac {2 \int -\frac {a (b c+3 a d)-b (b c-a d) x}{2 \sqrt {a+b x} (c+d x)^{5/2}}dx}{b^2 (b c-a d)}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 c \left (-\frac {\int \frac {a (b c+3 a d)-b (b c-a d) x}{\sqrt {a+b x} (c+d x)^{5/2}}dx}{b^2 (b c-a d)}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {2 c \left (-\frac {\frac {2 \sqrt {a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {b \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}}dx}{3 d (b c-a d)}}{b^2 (b c-a d)}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {2 c \left (-\frac {\frac {2 \sqrt {a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 d (c+d x)^{3/2} (b c-a d)}-\frac {2 b \sqrt {a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 d \sqrt {c+d x} (b c-a d)^2}}{b^2 (b c-a d)}-\frac {2 a^2}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}\right )}{b c-a d}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}\) |
(-2*x^3)/(3*(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (2*c*((-2*a^2)/ (b^2*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) - ((2*(b^2*c^2 + 3*a^2*d^2 )*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - (2*b*(b^2*c^2 - 6*a*b *c*d - 3*a^2*d^2)*Sqrt[a + b*x])/(3*d*(b*c - a*d)^2*Sqrt[c + d*x]))/(b^2*( b*c - a*d))))/(b*c - a*d)
3.9.7.3.1 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 + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 1])))
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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(-\frac {2 \left (-x^{3} a^{3} d^{3}+9 x^{3} a^{2} b c \,d^{2}+9 x^{3} a \,b^{2} c^{2} d -x^{3} b^{3} c^{3}+6 a^{3} c \,d^{2} x^{2}+36 a^{2} b \,c^{2} d \,x^{2}+6 a \,b^{2} c^{3} x^{2}+24 a^{3} c^{2} d x +24 a^{2} b \,c^{3} x +16 c^{3} a^{3}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}\) | \(141\) |
| gosper | \(-\frac {2 \left (-x^{3} a^{3} d^{3}+9 x^{3} a^{2} b c \,d^{2}+9 x^{3} a \,b^{2} c^{2} d -x^{3} b^{3} c^{3}+6 a^{3} c \,d^{2} x^{2}+36 a^{2} b \,c^{2} d \,x^{2}+6 a \,b^{2} c^{3} x^{2}+24 a^{3} c^{2} d x +24 a^{2} b \,c^{3} x +16 c^{3} a^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(182\) |
-2/3*(-a^3*d^3*x^3+9*a^2*b*c*d^2*x^3+9*a*b^2*c^2*d*x^3-b^3*c^3*x^3+6*a^3*c *d^2*x^2+36*a^2*b*c^2*d*x^2+6*a*b^2*c^3*x^2+24*a^3*c^2*d*x+24*a^2*b*c^3*x+ 16*a^3*c^3)/(a*d-b*c)^4/(b*x+a)^(3/2)/(d*x+c)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (167) = 334\).
Time = 0.59 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.37 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (16 \, a^{3} c^{3} - {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} + 6 \, {\left (a b^{2} c^{3} + 6 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} + 24 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
-2/3*(16*a^3*c^3 - (b^3*c^3 - 9*a*b^2*c^2*d - 9*a^2*b*c*d^2 + a^3*d^3)*x^3 + 6*(a*b^2*c^3 + 6*a^2*b*c^2*d + a^3*c*d^2)*x^2 + 24*(a^2*b*c^3 + a^3*c^2 *d)*x)*sqrt(b*x + a)*sqrt(d*x + c)/(a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4* b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4 + (b^6*c^4*d^2 - 4*a*b^5*c^3*d ^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d ^5 + a^5*b*d^6)*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3* b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*x)
\[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{3}}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
Exception generated. \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 771 vs. \(2 (167) = 334\).
Time = 0.55 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.08 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{7} c^{6} d {\left | b \right |} - 12 \, a b^{6} c^{5} d^{2} {\left | b \right |} + 30 \, a^{2} b^{5} c^{4} d^{3} {\left | b \right |} - 28 \, a^{3} b^{4} c^{3} d^{4} {\left | b \right |} + 9 \, a^{4} b^{3} c^{2} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} - \frac {9 \, {\left (a b^{7} c^{6} d {\left | b \right |} - 4 \, a^{2} b^{6} c^{5} d^{2} {\left | b \right |} + 6 \, a^{3} b^{5} c^{4} d^{3} {\left | b \right |} - 4 \, a^{4} b^{4} c^{3} d^{4} {\left | b \right |} + a^{5} b^{3} c^{2} d^{5} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (9 \, \sqrt {b d} a^{2} b^{5} c^{3} - 19 \, \sqrt {b d} a^{3} b^{4} c^{2} d + 11 \, \sqrt {b d} a^{4} b^{3} c d^{2} - \sqrt {b d} a^{5} b^{2} d^{3} - 18 \, \sqrt {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} a^{2} b^{3} c^{2} + 18 \, \sqrt {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} a^{3} b^{2} c d + 9 \, \sqrt {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 )}^{4} a^{2} b c - 3 \, \sqrt {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 )}^{4} a^{3} d\right )}}{3 \, {\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\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 )}^{3}} \]
2/3*sqrt(b*x + a)*((b^7*c^6*d*abs(b) - 12*a*b^6*c^5*d^2*abs(b) + 30*a^2*b^ 5*c^4*d^3*abs(b) - 28*a^3*b^4*c^3*d^4*abs(b) + 9*a^4*b^3*c^2*d^5*abs(b))*( b*x + a)/(b^9*c^7*d - 7*a*b^8*c^6*d^2 + 21*a^2*b^7*c^5*d^3 - 35*a^3*b^6*c^ 4*d^4 + 35*a^4*b^5*c^3*d^5 - 21*a^5*b^4*c^2*d^6 + 7*a^6*b^3*c*d^7 - a^7*b^ 2*d^8) - 9*(a*b^7*c^6*d*abs(b) - 4*a^2*b^6*c^5*d^2*abs(b) + 6*a^3*b^5*c^4* d^3*abs(b) - 4*a^4*b^4*c^3*d^4*abs(b) + a^5*b^3*c^2*d^5*abs(b))/(b^9*c^7*d - 7*a*b^8*c^6*d^2 + 21*a^2*b^7*c^5*d^3 - 35*a^3*b^6*c^4*d^4 + 35*a^4*b^5* c^3*d^5 - 21*a^5*b^4*c^2*d^6 + 7*a^6*b^3*c*d^7 - a^7*b^2*d^8))/(b^2*c + (b *x + a)*b*d - a*b*d)^(3/2) - 4/3*(9*sqrt(b*d)*a^2*b^5*c^3 - 19*sqrt(b*d)*a ^3*b^4*c^2*d + 11*sqrt(b*d)*a^4*b^3*c*d^2 - sqrt(b*d)*a^5*b^2*d^3 - 18*sqr t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a ^2*b^3*c^2 + 18*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^2*a^3*b^2*c*d + 9*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqr t(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2*b*c - 3*sqrt(b*d)*(sqrt(b*d)*sqrt( b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^3*d)/((b^3*c^3*abs(b) - 3*a*b^2*c^2*d*abs(b) + 3*a^2*b*c*d^2*abs(b) - a^3*d^3*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)^ 3)
Timed out. \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^3}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]