Integrand size = 22, antiderivative size = 352 \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {b (5 b c-3 a d)}{3 a^2 c (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {b \left (5 b^2 c^2-10 a b c d+a^2 d^2\right )}{a^3 c (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {d \left (15 b^3 c^3-35 a b^2 c^2 d+9 a^2 b c d^2-5 a^3 d^3\right ) \sqrt {a+b x}}{3 a^3 c^2 (b c-a d)^3 (c+d x)^{3/2}}-\frac {d \left (15 b^4 c^4-40 a b^3 c^3 d+18 a^2 b^2 c^2 d^2-40 a^3 b c d^3+15 a^4 d^4\right ) \sqrt {a+b x}}{3 a^3 c^3 (b c-a d)^4 \sqrt {c+d x}}+\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{7/2} c^{7/2}} \] Output:
-1/3*b*(-3*a*d+5*b*c)/a^2/c/(-a*d+b*c)/(b*x+a)^(3/2)/(d*x+c)^(3/2)-1/a/c/x /(b*x+a)^(3/2)/(d*x+c)^(3/2)-b*(a^2*d^2-10*a*b*c*d+5*b^2*c^2)/a^3/c/(-a*d+ b*c)^2/(b*x+a)^(1/2)/(d*x+c)^(3/2)-1/3*d*(-5*a^3*d^3+9*a^2*b*c*d^2-35*a*b^ 2*c^2*d+15*b^3*c^3)*(b*x+a)^(1/2)/a^3/c^2/(-a*d+b*c)^3/(d*x+c)^(3/2)-1/3*d *(15*a^4*d^4-40*a^3*b*c*d^3+18*a^2*b^2*c^2*d^2-40*a*b^3*c^3*d+15*b^4*c^4)* (b*x+a)^(1/2)/a^3/c^3/(-a*d+b*c)^4/(d*x+c)^(1/2)+5*(a*d+b*c)*arctanh(c^(1/ 2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)
Time = 0.42 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.93 \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {-15 b^6 c^4 x^2 (c+d x)^2-20 a b^5 c^3 x (c-2 d x) (c+d x)^2-3 a^2 b^4 c^2 (c+d x)^2 \left (c^2-18 c d x+6 d^2 x^2\right )-a^6 d^4 \left (3 c^2+20 c d x+15 d^2 x^2\right )+6 a^5 b d^3 \left (2 c^3+8 c^2 d x-5 d^3 x^3\right )-3 a^4 b^2 d^2 \left (6 c^4+4 c^3 d x-29 c^2 d^2 x^2-20 c d^3 x^3+5 d^4 x^4\right )+2 a^3 b^3 c d \left (6 c^4-6 c^3 d x-24 c^2 d^2 x^2+9 c d^3 x^3+20 d^4 x^4\right )}{3 a^3 c^3 (b c-a d)^4 x (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {5 (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{a^{7/2} c^{7/2}} \] Input:
Integrate[1/(x^2*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
Output:
(-15*b^6*c^4*x^2*(c + d*x)^2 - 20*a*b^5*c^3*x*(c - 2*d*x)*(c + d*x)^2 - 3* a^2*b^4*c^2*(c + d*x)^2*(c^2 - 18*c*d*x + 6*d^2*x^2) - a^6*d^4*(3*c^2 + 20 *c*d*x + 15*d^2*x^2) + 6*a^5*b*d^3*(2*c^3 + 8*c^2*d*x - 5*d^3*x^3) - 3*a^4 *b^2*d^2*(6*c^4 + 4*c^3*d*x - 29*c^2*d^2*x^2 - 20*c*d^3*x^3 + 5*d^4*x^4) + 2*a^3*b^3*c*d*(6*c^4 - 6*c^3*d*x - 24*c^2*d^2*x^2 + 9*c*d^3*x^3 + 20*d^4* x^4))/(3*a^3*c^3*(b*c - a*d)^4*x*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + (5*(b* c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(a^(7/2 )*c^(7/2))
Time = 0.55 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {114, 27, 169, 27, 169, 27, 169, 27, 169, 27, 104, 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.
\(\displaystyle \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {\int \frac {5 (b c+a d)+8 b d x}{2 x (a+b x)^{5/2} (c+d x)^{5/2}}dx}{a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {5 (b c+a d)+8 b d x}{x (a+b x)^{5/2} (c+d x)^{5/2}}dx}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {\frac {2 \int \frac {3 (5 (b c-a d) (b c+a d)+2 b d (5 b c-3 a d) x)}{2 x (a+b x)^{3/2} (c+d x)^{5/2}}dx}{3 a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\int \frac {5 (b c-a d) (b c+a d)+2 b d (5 b c-3 a d) x}{x (a+b x)^{3/2} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {\frac {\frac {2 \int \frac {5 (b c+a d) (b c-a d)^2+4 b d \left (5 b^2 c^2-10 a b d c+a^2 d^2\right ) x}{2 x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\frac {\int \frac {5 (b c+a d) (b c-a d)^2+4 b d \left (5 b^2 c^2-10 a b d c+a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)^{5/2}}dx}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {\frac {\frac {\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}-\frac {2 \int -\frac {15 (b c+a d) (b c-a d)^3+2 b d \left (15 b^3 c^3-35 a b^2 d c^2+9 a^2 b d^2 c-5 a^3 d^3\right ) x}{2 x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {15 (b c+a d) (b c-a d)^3+2 b d \left (15 b^3 c^3-35 a b^2 d c^2+9 a^2 b d^2 c-5 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)^{3/2}}dx}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle -\frac {\frac {\frac {\frac {\frac {2 d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {2 \int -\frac {15 (b c-a d)^4 (b c+a d)}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{c (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {\frac {\frac {\frac {15 (a d+b c) (b c-a d)^3 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{c}+\frac {2 d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle -\frac {\frac {\frac {\frac {\frac {30 (a d+b c) (b c-a d)^3 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{c}+\frac {2 d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}}{3 c (b c-a d)}+\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {\frac {\frac {2 b \left (a^2 d^2-10 a b c d+5 b^2 c^2\right )}{a \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)}+\frac {\frac {2 d \sqrt {a+b x} \left (-5 a^3 d^3+9 a^2 b c d^2-35 a b^2 c^2 d+15 b^3 c^3\right )}{3 c (c+d x)^{3/2} (b c-a d)}+\frac {\frac {2 d \sqrt {a+b x} \left (15 a^4 d^4-40 a^3 b c d^3+18 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+15 b^4 c^4\right )}{c \sqrt {c+d x} (b c-a d)}-\frac {30 (b c-a d)^3 (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} c^{3/2}}}{3 c (b c-a d)}}{a (b c-a d)}}{a (b c-a d)}+\frac {2 b (5 b c-3 a d)}{3 a (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}}{2 a c}-\frac {1}{a c x (a+b x)^{3/2} (c+d x)^{3/2}}\) |
Input:
Int[1/(x^2*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x]
Output:
-(1/(a*c*x*(a + b*x)^(3/2)*(c + d*x)^(3/2))) - ((2*b*(5*b*c - 3*a*d))/(3*a *(b*c - a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2)) + ((2*b*(5*b^2*c^2 - 10*a*b* c*d + a^2*d^2))/(a*(b*c - a*d)*Sqrt[a + b*x]*(c + d*x)^(3/2)) + ((2*d*(15* b^3*c^3 - 35*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*Sqrt[a + b*x])/(3*c* (b*c - a*d)*(c + d*x)^(3/2)) + ((2*d*(15*b^4*c^4 - 40*a*b^3*c^3*d + 18*a^2 *b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 15*a^4*d^4)*Sqrt[a + b*x])/(c*(b*c - a*d)* Sqrt[c + d*x]) - (30*(b*c - a*d)^3*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b *x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*c^(3/2)))/(3*c*(b*c - a*d)))/(a*(b *c - a*d)))/(a*(b*c - a*d)))/(2*a*c)
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_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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. \(2702\) vs. \(2(314)=628\).
Time = 0.33 (sec) , antiderivative size = 2703, normalized size of antiderivative = 7.68
Input:
int(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/6/c^3/a^3*(-45*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a *c)/x)*a^6*b*c^3*d^4*x+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^ (1/2)+2*a*c)/x)*a^5*b^2*c^4*d^3*x+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a )*(d*x+c))^(1/2)+2*a*c)/x)*a^4*b^3*c^5*d^2*x-45*ln((a*d*x+b*c*x+2*(a*c)^(1 /2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*b^4*c^6*d*x-96*(a*c)^(1/2)*((b*x +a)*(d*x+c))^(1/2)*a^3*b^3*c^3*d^3*x^2+96*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 /2)*a^5*b*c^2*d^4*x-24*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c^3*d^3 *x-24*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b^3*c^4*d^2*x+96*(a*c)^(1/2) *((b*x+a)*(d*x+c))^(1/2)*a^2*b^4*c^5*d*x-60*(a*c)^(1/2)*((b*x+a)*(d*x+c))^ (1/2)*a^5*b*d^6*x^3-60*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^6*c^5*d*x^3+2 4*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^5*b*c^3*d^3-36*(a*c)^(1/2)*((b*x+a )*(d*x+c))^(1/2)*a^4*b^2*c^4*d^2+24*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^ 3*b^3*c^5*d-36*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b^4*c^2*d^4*x^4+120 *(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*b^2*c*d^5*x^3+36*(a*c)^(1/2)*((b* x+a)*(d*x+c))^(1/2)*a^3*b^3*c^2*d^4*x^3+36*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)*a^2*b^4*c^3*d^3*x^3+120*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b^5*c^4 *d^2*x^3-60*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x )*a^6*b*c^2*d^5*x^2+105*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1 /2)+2*a*c)/x)*a^3*b^4*c^4*d^3*x^3-135*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+ a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b^5*c^5*d^2*x^3+15*ln((a*d*x+b*c*x+2*(a...
Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (314) = 628\).
Time = 5.21 (sec) , antiderivative size = 2506, normalized size of antiderivative = 7.12 \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="fricas")
Output:
[1/12*(15*((b^7*c^5*d^2 - 3*a*b^6*c^4*d^3 + 2*a^2*b^5*c^3*d^4 + 2*a^3*b^4* c^2*d^5 - 3*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^5 + 2*(b^7*c^6*d - 2*a*b^6*c^5* d^2 - a^2*b^5*c^4*d^3 + 4*a^3*b^4*c^3*d^4 - a^4*b^3*c^2*d^5 - 2*a^5*b^2*c* d^6 + a^6*b*d^7)*x^4 + (b^7*c^7 + a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 7*a^3* b^4*c^4*d^3 + 7*a^4*b^3*c^3*d^4 - 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^ 7)*x^3 + 2*(a*b^6*c^7 - 2*a^2*b^5*c^6*d - a^3*b^4*c^5*d^2 + 4*a^4*b^3*c^4* d^3 - a^5*b^2*c^3*d^4 - 2*a^6*b*c^2*d^5 + a^7*c*d^6)*x^2 + (a^2*b^5*c^7 - 3*a^3*b^4*c^6*d + 2*a^4*b^3*c^5*d^2 + 2*a^5*b^2*c^4*d^3 - 3*a^6*b*c^3*d^4 + a^7*c^2*d^5)*x)*sqrt(a*c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^ 2)*x^2 + 4*(2*a*c + (b*c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) - 4*(3*a^3*b^4*c^7 - 12*a^4*b^3*c^6*d + 18* a^5*b^2*c^5*d^2 - 12*a^6*b*c^4*d^3 + 3*a^7*c^3*d^4 + (15*a*b^6*c^5*d^2 - 4 0*a^2*b^5*c^4*d^3 + 18*a^3*b^4*c^3*d^4 - 40*a^4*b^3*c^2*d^5 + 15*a^5*b^2*c *d^6)*x^4 + 6*(5*a*b^6*c^6*d - 10*a^2*b^5*c^5*d^2 - 3*a^3*b^4*c^4*d^3 - 3* a^4*b^3*c^3*d^4 - 10*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6)*x^3 + 3*(5*a*b^6*c^7 - 29*a^3*b^4*c^5*d^2 + 16*a^4*b^3*c^4*d^3 - 29*a^5*b^2*c^3*d^4 + 5*a^7*c* d^6)*x^2 + 4*(5*a^2*b^5*c^7 - 12*a^3*b^4*c^6*d + 3*a^4*b^3*c^5*d^2 + 3*a^5 *b^2*c^4*d^3 - 12*a^6*b*c^3*d^4 + 5*a^7*c^2*d^5)*x)*sqrt(b*x + a)*sqrt(d*x + c))/((a^4*b^6*c^8*d^2 - 4*a^5*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b ^3*c^5*d^5 + a^8*b^2*c^4*d^6)*x^5 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^...
\[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{2} \left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/x**2/(b*x+a)**(5/2)/(d*x+c)**(5/2),x)
Output:
Integral(1/(x**2*(a + b*x)**(5/2)*(c + d*x)**(5/2)), x)
Exception generated. \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="maxima")
Output:
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. 1272 vs. \(2 (314) = 628\).
Time = 2.28 (sec) , antiderivative size = 1272, normalized size of antiderivative = 3.61 \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x, algorithm="giac")
Output:
2/3*sqrt(b*x + a)*(2*(7*b^7*c^7*d^6*abs(b) - 24*a*b^6*c^6*d^7*abs(b) + 30* a^2*b^5*c^5*d^8*abs(b) - 16*a^3*b^4*c^4*d^9*abs(b) + 3*a^4*b^3*c^3*d^10*ab s(b))*(b*x + a)/(b^9*c^13*d - 7*a*b^8*c^12*d^2 + 21*a^2*b^7*c^11*d^3 - 35* a^3*b^6*c^10*d^4 + 35*a^4*b^5*c^9*d^5 - 21*a^5*b^4*c^8*d^6 + 7*a^6*b^3*c^7 *d^7 - a^7*b^2*c^6*d^8) + 3*(5*b^8*c^8*d^5*abs(b) - 22*a*b^7*c^7*d^6*abs(b ) + 38*a^2*b^6*c^6*d^7*abs(b) - 32*a^3*b^5*c^5*d^8*abs(b) + 13*a^4*b^4*c^4 *d^9*abs(b) - 2*a^5*b^3*c^3*d^10*abs(b))/(b^9*c^13*d - 7*a*b^8*c^12*d^2 + 21*a^2*b^7*c^11*d^3 - 35*a^3*b^6*c^10*d^4 + 35*a^4*b^5*c^9*d^5 - 21*a^5*b^ 4*c^8*d^6 + 7*a^6*b^3*c^7*d^7 - a^7*b^2*c^6*d^8))/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 8/3*(3*sqrt(b*d)*b^10*c^3 - 13*sqrt(b*d)*a*b^9*c^2*d + 17* sqrt(b*d)*a^2*b^8*c*d^2 - 7*sqrt(b*d)*a^3*b^7*d^3 - 6*sqrt(b*d)*(sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^8*c^2 + 21*sqrt( b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b ^7*c*d - 15*sqrt(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^6*d^2 + 3*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^ 2*c + (b*x + a)*b*d - a*b*d))^4*b^6*c - 6*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^5*d)/((a^3*b^3*c^3*abs(b) - 3*a^4*b^2*c^2*d*abs(b) + 3*a^5*b*c*d^2*abs(b) - a^6*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) + 5*(sqrt(b*d)*b^3*c + sqrt(b*d)*a*b^2*d)*arctan(-1/2*(b^2*c + a*b...
Timed out. \[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:
int(1/(x^2*(a + b*x)^(5/2)*(c + d*x)^(5/2)),x)
Output:
int(1/(x^2*(a + b*x)^(5/2)*(c + d*x)^(5/2)), x)
\[ \int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{x^{2} \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {5}{2}}}d x \] Input:
int(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)
Output:
int(1/x^2/(b*x+a)^(5/2)/(d*x+c)^(5/2),x)