Integrand size = 28, antiderivative size = 367 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {1}{3 a c (e x)^{5/2} \sqrt {a x+b x^2}}+\frac {7 b c+6 a d}{12 a^2 c^2 e (e x)^{3/2} \sqrt {a x+b x^2}}-\frac {35 b^2 c^2+30 a b c d+24 a^2 d^2}{24 a^3 c^3 e^2 \sqrt {e x} \sqrt {a x+b x^2}}-\frac {b \left (35 b^3 c^3-5 a b^2 c^2 d-6 a^2 b c d^2-8 a^3 d^3\right ) \sqrt {e x}}{8 a^4 c^3 (b c-a d) e^3 \sqrt {a x+b x^2}}-\frac {2 d^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {e} \sqrt {a x+b x^2}}{\sqrt {b c-a d} \sqrt {e x}}\right )}{c^4 (b c-a d)^{3/2} e^{5/2}}+\frac {\left (35 b^3 c^3+30 a b^2 c^2 d+24 a^2 b c d^2+16 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {e x}}\right )}{8 a^{9/2} c^4 e^{5/2}} \] Output:
-1/3/a/c/(e*x)^(5/2)/(b*x^2+a*x)^(1/2)+1/12*(6*a*d+7*b*c)/a^2/c^2/e/(e*x)^ (3/2)/(b*x^2+a*x)^(1/2)-1/24*(24*a^2*d^2+30*a*b*c*d+35*b^2*c^2)/a^3/c^3/e^ 2/(e*x)^(1/2)/(b*x^2+a*x)^(1/2)-1/8*b*(-8*a^3*d^3-6*a^2*b*c*d^2-5*a*b^2*c^ 2*d+35*b^3*c^3)*(e*x)^(1/2)/a^4/c^3/(-a*d+b*c)/e^3/(b*x^2+a*x)^(1/2)-2*d^( 9/2)*arctan(d^(1/2)*e^(1/2)*(b*x^2+a*x)^(1/2)/(-a*d+b*c)^(1/2)/(e*x)^(1/2) )/c^4/(-a*d+b*c)^(3/2)/e^(5/2)+1/8*(16*a^3*d^3+24*a^2*b*c*d^2+30*a*b^2*c^2 *d+35*b^3*c^3)*arctanh(e^(1/2)*(b*x^2+a*x)^(1/2)/a^(1/2)/(e*x)^(1/2))/a^(9 /2)/c^4/e^(5/2)
Time = 0.77 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {\sqrt {a} \left (c \sqrt {b c-a d} \left (105 b^4 c^3 x^3+5 a b^3 c^2 x^2 (7 c-3 d x)-4 a^4 d \left (2 c^2-3 c d x+6 d^2 x^2\right )-a^2 b^2 c x \left (14 c^2+5 c d x+18 d^2 x^2\right )+2 a^3 b \left (4 c^3+c^2 d x-3 c d^2 x^2-12 d^3 x^3\right )\right )+48 a^4 d^{9/2} x^3 \sqrt {a+b x} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )\right )-3 \sqrt {b c-a d} \left (35 b^4 c^4-5 a b^3 c^3 d-6 a^2 b^2 c^2 d^2-8 a^3 b c d^3-16 a^4 d^4\right ) x^3 \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{24 a^{9/2} c^4 (b c-a d)^{3/2} (e x)^{5/2} \sqrt {x (a+b x)}} \] Input:
Integrate[1/((e*x)^(5/2)*(c + d*x)*(a*x + b*x^2)^(3/2)),x]
Output:
-1/24*(Sqrt[a]*(c*Sqrt[b*c - a*d]*(105*b^4*c^3*x^3 + 5*a*b^3*c^2*x^2*(7*c - 3*d*x) - 4*a^4*d*(2*c^2 - 3*c*d*x + 6*d^2*x^2) - a^2*b^2*c*x*(14*c^2 + 5 *c*d*x + 18*d^2*x^2) + 2*a^3*b*(4*c^3 + c^2*d*x - 3*c*d^2*x^2 - 12*d^3*x^3 )) + 48*a^4*d^(9/2)*x^3*Sqrt[a + b*x]*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[ b*c - a*d]]) - 3*Sqrt[b*c - a*d]*(35*b^4*c^4 - 5*a*b^3*c^3*d - 6*a^2*b^2*c ^2*d^2 - 8*a^3*b*c*d^3 - 16*a^4*d^4)*x^3*Sqrt[a + b*x]*ArcTanh[Sqrt[a + b* x]/Sqrt[a]])/(a^(9/2)*c^4*(b*c - a*d)^(3/2)*(e*x)^(5/2)*Sqrt[x*(a + b*x)])
Time = 0.94 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.98, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {1261, 114, 27, 168, 27, 168, 27, 169, 27, 174, 73, 218, 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.
| \(\displaystyle \int \frac {1}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2} (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \int \frac {1}{x^4 (a+b x)^{3/2} (c+d x)}dx}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {\int \frac {7 b c+6 a d+7 b d x}{2 x^3 (a+b x)^{3/2} (c+d x)}dx}{3 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {\int \frac {7 b c+6 a d+7 b d x}{x^3 (a+b x)^{3/2} (c+d x)}dx}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {\int \frac {35 b^2 c^2+30 a b d c+24 a^2 d^2+5 b d (7 b c+6 a d) x}{2 x^2 (a+b x)^{3/2} (c+d x)}dx}{2 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {\int \frac {35 b^2 c^2+30 a b d c+24 a^2 d^2+5 b d (7 b c+6 a d) x}{x^2 (a+b x)^{3/2} (c+d x)}dx}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {\int \frac {3 \left (35 b^3 c^3+30 a b^2 d c^2+24 a^2 b d^2 c+16 a^3 d^3+b d \left (35 b^2 c^2+30 a b d c+24 a^2 d^2\right ) x\right )}{2 x (a+b x)^{3/2} (c+d x)}dx}{a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \int \frac {35 b^3 c^3+30 a b^2 d c^2+24 a^2 b d^2 c+16 a^3 d^3+b d \left (35 b^2 c^2+30 a b d c+24 a^2 d^2\right ) x}{x (a+b x)^{3/2} (c+d x)}dx}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {2 \int \frac {(b c-a d) \left (35 b^3 c^3+30 a b^2 d c^2+24 a^2 b d^2 c+16 a^3 d^3\right )+b d \left (35 b^3 c^3-5 a b^2 d c^2-6 a^2 b d^2 c-8 a^3 d^3\right ) x}{2 x \sqrt {a+b x} (c+d x)}dx}{a (b c-a d)}+\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {\int \frac {(b c-a d) \left (35 b^3 c^3+30 a b^2 d c^2+24 a^2 b d^2 c+16 a^3 d^3\right )+b d \left (35 b^3 c^3-5 a b^2 d c^2-6 a^2 b d^2 c-8 a^3 d^3\right ) x}{x \sqrt {a+b x} (c+d x)}dx}{a (b c-a d)}+\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {16 a^4 d^5 \int \frac {1}{\sqrt {a+b x} (c+d x)}dx}{c}+\frac {(b c-a d) \left (16 a^3 d^3+24 a^2 b c d^2+30 a b^2 c^2 d+35 b^3 c^3\right ) \int \frac {1}{x \sqrt {a+b x}}dx}{c}}{a (b c-a d)}+\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {32 a^4 d^5 \int \frac {1}{c-\frac {a d}{b}+\frac {d (a+b x)}{b}}d\sqrt {a+b x}}{b c}+\frac {2 (b c-a d) \left (16 a^3 d^3+24 a^2 b c d^2+30 a b^2 c^2 d+35 b^3 c^3\right ) \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}}{a (b c-a d)}+\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {\frac {2 (b c-a d) \left (16 a^3 d^3+24 a^2 b c d^2+30 a b^2 c^2 d+35 b^3 c^3\right ) \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}+\frac {32 a^4 d^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}}{a (b c-a d)}+\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {x^4 (a+b x)^{3/2} \left (-\frac {-\frac {-\frac {3 \left (\frac {2 b \left (-8 a^3 d^3-6 a^2 b c d^2-5 a b^2 c^2 d+35 b^3 c^3\right )}{a \sqrt {a+b x} (b c-a d)}+\frac {\frac {32 a^4 d^{9/2} \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{c \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) (b c-a d) \left (16 a^3 d^3+24 a^2 b c d^2+30 a b^2 c^2 d+35 b^3 c^3\right )}{\sqrt {a} c}}{a (b c-a d)}\right )}{2 a c}-\frac {\frac {35 b^2 c}{a}+\frac {24 a d^2}{c}+30 b d}{x \sqrt {a+b x}}}{4 a c}-\frac {6 a d+7 b c}{2 a c x^2 \sqrt {a+b x}}}{6 a c}-\frac {1}{3 a c x^3 \sqrt {a+b x}}\right )}{(e x)^{5/2} \left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[1/((e*x)^(5/2)*(c + d*x)*(a*x + b*x^2)^(3/2)),x]
Output:
(x^4*(a + b*x)^(3/2)*(-1/3*1/(a*c*x^3*Sqrt[a + b*x]) - (-1/2*(7*b*c + 6*a* d)/(a*c*x^2*Sqrt[a + b*x]) - (-(((35*b^2*c)/a + 30*b*d + (24*a*d^2)/c)/(x* Sqrt[a + b*x])) - (3*((2*b*(35*b^3*c^3 - 5*a*b^2*c^2*d - 6*a^2*b*c*d^2 - 8 *a^3*d^3))/(a*(b*c - a*d)*Sqrt[a + b*x]) + ((32*a^4*d^(9/2)*ArcTan[(Sqrt[d ]*Sqrt[a + b*x])/Sqrt[b*c - a*d]])/(c*Sqrt[b*c - a*d]) - (2*(b*c - a*d)*(3 5*b^3*c^3 + 30*a*b^2*c^2*d + 24*a^2*b*c*d^2 + 16*a^3*d^3)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(Sqrt[a]*c))/(a*(b*c - a*d))))/(2*a*c))/(4*a*c))/(6*a*c))) /((e*x)^(5/2)*(a*x + b*x^2)^(3/2))
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] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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] && ILtQ[m, -1]
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[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) ^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, n}, x] && !IGtQ[n, 0]
Time = 0.85 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {\left (b x +a \right ) \left (24 a^{2} d^{2} x^{2}+42 a b c d \,x^{2}+57 b^{2} c^{2} x^{2}-12 a^{2} c d x -22 a b \,c^{2} x +8 a^{2} c^{2}\right )}{24 a^{4} c^{3} x^{2} e^{2} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}-\frac {b \left (-\frac {\left (16 a^{3} d^{3}+24 a^{2} b c \,d^{2}+30 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b e x +a e}}{\sqrt {a e}}\right )}{b c \sqrt {a e}}-\frac {16 b^{3} c^{3}}{\left (a d -b c \right ) \sqrt {b e x +a e}}+\frac {16 a^{4} d^{5} \operatorname {arctanh}\left (\frac {d \sqrt {b e x +a e}}{\sqrt {e \left (a d -b c \right ) d}}\right )}{\left (a d -b c \right ) b c \sqrt {e \left (a d -b c \right ) d}}\right ) \sqrt {\left (b x +a \right ) e}\, x}{8 c^{3} a^{4} e^{2} \sqrt {e x}\, \sqrt {x \left (b x +a \right )}}\) | \(286\) |
| default | \(-\frac {\sqrt {x \left (b x +a \right )}\, \left (48 \sqrt {\left (b x +a \right ) e}\, \sqrt {a e}\, \operatorname {arctanh}\left (\frac {d \sqrt {\left (b x +a \right ) e}}{\sqrt {e \left (a d -b c \right ) d}}\right ) a^{4} d^{5} x^{3}-48 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, a^{4} d^{4} x^{3}-24 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, a^{3} b c \,d^{3} x^{3}-18 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, a^{2} b^{2} c^{2} d^{2} x^{3}-15 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, a \,b^{3} c^{3} d \,x^{3}+105 \sqrt {\left (b x +a \right ) e}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (b x +a \right ) e}}{\sqrt {a e}}\right ) \sqrt {e \left (a d -b c \right ) d}\, b^{4} c^{4} x^{3}+24 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{3} b c \,d^{3} x^{3}+18 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{2} b^{2} c^{2} d^{2} x^{3}+15 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a \,b^{3} c^{3} d \,x^{3}-105 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, b^{4} c^{4} x^{3}+24 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{4} c \,d^{3} x^{2}+6 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{3} b \,c^{2} d^{2} x^{2}+5 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{2} b^{2} c^{3} d \,x^{2}-35 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a \,b^{3} c^{4} x^{2}-12 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{4} c^{2} d^{2} x -2 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{3} b \,c^{3} d x +14 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{2} b^{2} c^{4} x +8 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{4} c^{3} d -8 \sqrt {a e}\, \sqrt {e \left (a d -b c \right ) d}\, a^{3} b \,c^{4}\right )}{24 e^{2} x^{3} \sqrt {e x}\, \left (b x +a \right ) a^{4} c^{4} \sqrt {a e}\, \left (a d -b c \right ) \sqrt {e \left (a d -b c \right ) d}}\) | \(765\) |
Input:
int(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/24*(b*x+a)*(24*a^2*d^2*x^2+42*a*b*c*d*x^2+57*b^2*c^2*x^2-12*a^2*c*d*x-2 2*a*b*c^2*x+8*a^2*c^2)/a^4/c^3/x^2/e^2/(e*x)^(1/2)/(x*(b*x+a))^(1/2)-1/8/c ^3/a^4*b*(-(16*a^3*d^3+24*a^2*b*c*d^2+30*a*b^2*c^2*d+35*b^3*c^3)/b/c/(a*e) ^(1/2)*arctanh((b*e*x+a*e)^(1/2)/(a*e)^(1/2))-16*b^3*c^3/(a*d-b*c)/(b*e*x+ a*e)^(1/2)+16/(a*d-b*c)*a^4*d^5/b/c/(e*(a*d-b*c)*d)^(1/2)*arctanh(d*(b*e*x +a*e)^(1/2)/(e*(a*d-b*c)*d)^(1/2)))/e^2*((b*x+a)*e)^(1/2)/(e*x)^(1/2)/(x*( b*x+a))^(1/2)*x
Time = 1.13 (sec) , antiderivative size = 2013, normalized size of antiderivative = 5.49 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="fricas")
Output:
[-1/48*(48*(a^5*b*d^4*e*x^5 + a^6*d^4*e*x^4)*sqrt(-d/((b*c - a*d)*e))*log( -(b*d*x^2 + 2*sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(e*x)*sqrt(-d/((b*c - a*d) *e)) - (b*c - 2*a*d)*x)/(d*x^2 + c*x)) - 3*((35*b^5*c^4 - 5*a*b^4*c^3*d - 6*a^2*b^3*c^2*d^2 - 8*a^3*b^2*c*d^3 - 16*a^4*b*d^4)*x^5 + (35*a*b^4*c^4 - 5*a^2*b^3*c^3*d - 6*a^3*b^2*c^2*d^2 - 8*a^4*b*c*d^3 - 16*a^5*d^4)*x^4)*sqr t(a*e)*log(-(b*e*x^2 + 2*a*e*x + 2*sqrt(b*x^2 + a*x)*sqrt(a*e)*sqrt(e*x))/ x^2) + 2*(8*a^4*b*c^4 - 8*a^5*c^3*d + 3*(35*a*b^4*c^4 - 5*a^2*b^3*c^3*d - 6*a^3*b^2*c^2*d^2 - 8*a^4*b*c*d^3)*x^3 + (35*a^2*b^3*c^4 - 5*a^3*b^2*c^3*d - 6*a^4*b*c^2*d^2 - 24*a^5*c*d^3)*x^2 - 2*(7*a^3*b^2*c^4 - a^4*b*c^3*d - 6*a^5*c^2*d^2)*x)*sqrt(b*x^2 + a*x)*sqrt(e*x))/((a^5*b^2*c^5 - a^6*b*c^4*d )*e^3*x^5 + (a^6*b*c^5 - a^7*c^4*d)*e^3*x^4), -1/24*(3*((35*b^5*c^4 - 5*a* b^4*c^3*d - 6*a^2*b^3*c^2*d^2 - 8*a^3*b^2*c*d^3 - 16*a^4*b*d^4)*x^5 + (35* a*b^4*c^4 - 5*a^2*b^3*c^3*d - 6*a^3*b^2*c^2*d^2 - 8*a^4*b*c*d^3 - 16*a^5*d ^4)*x^4)*sqrt(-a*e)*arctan(sqrt(b*x^2 + a*x)*sqrt(-a*e)*sqrt(e*x)/(a*e*x)) + 24*(a^5*b*d^4*e*x^5 + a^6*d^4*e*x^4)*sqrt(-d/((b*c - a*d)*e))*log(-(b*d *x^2 + 2*sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(e*x)*sqrt(-d/((b*c - a*d)*e)) - (b*c - 2*a*d)*x)/(d*x^2 + c*x)) + (8*a^4*b*c^4 - 8*a^5*c^3*d + 3*(35*a*b ^4*c^4 - 5*a^2*b^3*c^3*d - 6*a^3*b^2*c^2*d^2 - 8*a^4*b*c*d^3)*x^3 + (35*a^ 2*b^3*c^4 - 5*a^3*b^2*c^3*d - 6*a^4*b*c^2*d^2 - 24*a^5*c*d^3)*x^2 - 2*(7*a ^3*b^2*c^4 - a^4*b*c^3*d - 6*a^5*c^2*d^2)*x)*sqrt(b*x^2 + a*x)*sqrt(e*x...
\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (e x\right )^{\frac {5}{2}} \left (x \left (a + b x\right )\right )^{\frac {3}{2}} \left (c + d x\right )}\, dx \] Input:
integrate(1/(e*x)**(5/2)/(d*x+c)/(b*x**2+a*x)**(3/2),x)
Output:
Integral(1/((e*x)**(5/2)*(x*(a + b*x))**(3/2)*(c + d*x)), x)
\[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} {\left (d x + c\right )} \left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^2 + a*x)^(3/2)*(d*x + c)*(e*x)^(5/2)), x)
Time = 0.65 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=-\frac {1}{24} \, {\left (\frac {48 \, d^{5} \arctan \left (\frac {\sqrt {b e x + a e} d}{\sqrt {b c d e - a d^{2} e}}\right )}{{\left (b c^{5} e^{5} {\left | e \right |} - a c^{4} d e^{5} {\left | e \right |}\right )} \sqrt {b c d e - a d^{2} e}} + \frac {48 \, b^{4}}{{\left (a^{4} b c e^{5} {\left | e \right |} - a^{5} d e^{5} {\left | e \right |}\right )} \sqrt {b e x + a e}} + \frac {3 \, {\left (35 \, b^{3} c^{3} + 30 \, a b^{2} c^{2} d + 24 \, a^{2} b c d^{2} + 16 \, a^{3} d^{3}\right )} \arctan \left (\frac {\sqrt {b e x + a e}}{\sqrt {-a e}}\right )}{\sqrt {-a e} a^{4} c^{4} e^{5} {\left | e \right |}} + \frac {87 \, \sqrt {b e x + a e} a^{2} b^{3} c^{2} e^{2} + 54 \, \sqrt {b e x + a e} a^{3} b^{2} c d e^{2} + 24 \, \sqrt {b e x + a e} a^{4} b d^{2} e^{2} - 136 \, {\left (b e x + a e\right )}^{\frac {3}{2}} a b^{3} c^{2} e - 96 \, {\left (b e x + a e\right )}^{\frac {3}{2}} a^{2} b^{2} c d e - 48 \, {\left (b e x + a e\right )}^{\frac {3}{2}} a^{3} b d^{2} e + 57 \, {\left (b e x + a e\right )}^{\frac {5}{2}} b^{3} c^{2} + 42 \, {\left (b e x + a e\right )}^{\frac {5}{2}} a b^{2} c d + 24 \, {\left (b e x + a e\right )}^{\frac {5}{2}} a^{2} b d^{2}}{a^{4} b^{3} c^{3} e^{8} x^{3} {\left | e \right |}}\right )} e^{4} \] Input:
integrate(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x, algorithm="giac")
Output:
-1/24*(48*d^5*arctan(sqrt(b*e*x + a*e)*d/sqrt(b*c*d*e - a*d^2*e))/((b*c^5* e^5*abs(e) - a*c^4*d*e^5*abs(e))*sqrt(b*c*d*e - a*d^2*e)) + 48*b^4/((a^4*b *c*e^5*abs(e) - a^5*d*e^5*abs(e))*sqrt(b*e*x + a*e)) + 3*(35*b^3*c^3 + 30* a*b^2*c^2*d + 24*a^2*b*c*d^2 + 16*a^3*d^3)*arctan(sqrt(b*e*x + a*e)/sqrt(- a*e))/(sqrt(-a*e)*a^4*c^4*e^5*abs(e)) + (87*sqrt(b*e*x + a*e)*a^2*b^3*c^2* e^2 + 54*sqrt(b*e*x + a*e)*a^3*b^2*c*d*e^2 + 24*sqrt(b*e*x + a*e)*a^4*b*d^ 2*e^2 - 136*(b*e*x + a*e)^(3/2)*a*b^3*c^2*e - 96*(b*e*x + a*e)^(3/2)*a^2*b ^2*c*d*e - 48*(b*e*x + a*e)^(3/2)*a^3*b*d^2*e + 57*(b*e*x + a*e)^(5/2)*b^3 *c^2 + 42*(b*e*x + a*e)^(5/2)*a*b^2*c*d + 24*(b*e*x + a*e)^(5/2)*a^2*b*d^2 )/(a^4*b^3*c^3*e^8*x^3*abs(e)))*e^4
Timed out. \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\,x\right )}^{3/2}\,{\left (e\,x\right )}^{5/2}\,\left (c+d\,x\right )} \,d x \] Input:
int(1/((a*x + b*x^2)^(3/2)*(e*x)^(5/2)*(c + d*x)),x)
Output:
int(1/((a*x + b*x^2)^(3/2)*(e*x)^(5/2)*(c + d*x)), x)
Time = 0.22 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(e x)^{5/2} (c+d x) \left (a x+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {e}\, \left (-48 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{5} d^{5} x^{3}-105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{5} c^{5} x^{3}+48 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{5} d^{5} x^{3}+105 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{5} c^{5} x^{3}-16 a^{6} c^{3} d^{2}-16 a^{4} b^{2} c^{5}-96 \sqrt {d}\, \sqrt {b x +a}\, \sqrt {-a d +b c}\, \mathit {atan} \left (\frac {\sqrt {b x +a}\, d}{\sqrt {d}\, \sqrt {-a d +b c}}\right ) a^{5} d^{4} x^{3}+24 a^{6} c^{2} d^{3} x -48 a^{6} c \,d^{4} x^{2}+32 a^{5} b \,c^{4} d +28 a^{3} b^{3} c^{5} x -70 a^{2} b^{4} c^{5} x^{2}-210 a \,b^{5} c^{5} x^{3}+24 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{4} b c \,d^{4} x^{3}+6 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{3} b^{2} c^{2} d^{3} x^{3}+3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} b^{3} c^{3} d^{2} x^{3}+120 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{4} c^{4} d \,x^{3}-24 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{4} b c \,d^{4} x^{3}-6 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{3} b^{2} c^{2} d^{3} x^{3}-3 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} b^{3} c^{3} d^{2} x^{3}-120 \sqrt {a}\, \sqrt {b x +a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{4} c^{4} d \,x^{3}-20 a^{5} b \,c^{3} d^{2} x +36 a^{5} b \,c^{2} d^{3} x^{2}-48 a^{5} b c \,d^{4} x^{3}-32 a^{4} b^{2} c^{4} d x +2 a^{4} b^{2} c^{3} d^{2} x^{2}+12 a^{4} b^{2} c^{2} d^{3} x^{3}+80 a^{3} b^{3} c^{4} d \,x^{2}+6 a^{3} b^{3} c^{3} d^{2} x^{3}+240 a^{2} b^{4} c^{4} d \,x^{3}\right )}{48 \sqrt {b x +a}\, a^{5} c^{4} e^{3} x^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )} \] Input:
int(1/(e*x)^(5/2)/(d*x+c)/(b*x^2+a*x)^(3/2),x)
Output:
(sqrt(e)*( - 96*sqrt(d)*sqrt(a + b*x)*sqrt( - a*d + b*c)*atan((sqrt(a + b* x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a**5*d**4*x**3 - 48*sqrt(a)*sqrt(a + b *x)*log(sqrt(a + b*x) - sqrt(a))*a**5*d**5*x**3 + 24*sqrt(a)*sqrt(a + b*x) *log(sqrt(a + b*x) - sqrt(a))*a**4*b*c*d**4*x**3 + 6*sqrt(a)*sqrt(a + b*x) *log(sqrt(a + b*x) - sqrt(a))*a**3*b**2*c**2*d**3*x**3 + 3*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a**2*b**3*c**3*d**2*x**3 + 120*sqrt(a) *sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*a*b**4*c**4*d*x**3 - 105*sqrt( a)*sqrt(a + b*x)*log(sqrt(a + b*x) - sqrt(a))*b**5*c**5*x**3 + 48*sqrt(a)* sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**5*d**5*x**3 - 24*sqrt(a)*sqr t(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**4*b*c*d**4*x**3 - 6*sqrt(a)*sqr t(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**3*b**2*c**2*d**3*x**3 - 3*sqrt( a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a**2*b**3*c**3*d**2*x**3 - 1 20*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*a*b**4*c**4*d*x**3 + 105*sqrt(a)*sqrt(a + b*x)*log(sqrt(a + b*x) + sqrt(a))*b**5*c**5*x**3 - 1 6*a**6*c**3*d**2 + 24*a**6*c**2*d**3*x - 48*a**6*c*d**4*x**2 + 32*a**5*b*c **4*d - 20*a**5*b*c**3*d**2*x + 36*a**5*b*c**2*d**3*x**2 - 48*a**5*b*c*d** 4*x**3 - 16*a**4*b**2*c**5 - 32*a**4*b**2*c**4*d*x + 2*a**4*b**2*c**3*d**2 *x**2 + 12*a**4*b**2*c**2*d**3*x**3 + 28*a**3*b**3*c**5*x + 80*a**3*b**3*c **4*d*x**2 + 6*a**3*b**3*c**3*d**2*x**3 - 70*a**2*b**4*c**5*x**2 + 240*a** 2*b**4*c**4*d*x**3 - 210*a*b**5*c**5*x**3))/(48*sqrt(a + b*x)*a**5*c**4...