Integrand size = 24, antiderivative size = 220 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=-\frac {2 a (10 b c-7 a d) \sqrt {a x+b x^2}}{35 c^2 x^3}-\frac {2 \left (45 b^2 c^2-77 a b c d+35 a^2 d^2\right ) \sqrt {a x+b x^2}}{105 c^3 x^2}-\frac {2 \left (15 b^3 c^3-161 a b^2 c^2 d+245 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a x+b x^2}}{105 a c^4 x}-\frac {2 a \left (a x+b x^2\right )^{3/2}}{7 c x^5}-\frac {2 d (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{9/2}} \] Output:
-2/35*a*(-7*a*d+10*b*c)*(b*x^2+a*x)^(1/2)/c^2/x^3-2/105*(35*a^2*d^2-77*a*b *c*d+45*b^2*c^2)*(b*x^2+a*x)^(1/2)/c^3/x^2-2/105*(-105*a^3*d^3+245*a^2*b*c *d^2-161*a*b^2*c^2*d+15*b^3*c^3)*(b*x^2+a*x)^(1/2)/a/c^4/x-2/7*a*(b*x^2+a* x)^(3/2)/c/x^5-2*d*(-a*d+b*c)^(5/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b* x^2+a*x)^(1/2))/c^(9/2)
Time = 0.76 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\frac {2 (x (a+b x))^{5/2} \left (-\frac {\sqrt {c} \left (15 b^3 c^3 x^3+a b^2 c^2 x^2 (45 c-161 d x)+a^2 b c x \left (45 c^2-77 c d x+245 d^2 x^2\right )+a^3 \left (15 c^3-21 c^2 d x+35 c d^2 x^2-105 d^3 x^3\right )\right )}{a (a+b x)^2}-\frac {105 d (-b c+a d)^{5/2} x^{7/2} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )}{(a+b x)^{5/2}}\right )}{105 c^{9/2} x^6} \] Input:
Integrate[(a*x + b*x^2)^(5/2)/(x^7*(c + d*x)),x]
Output:
(2*(x*(a + b*x))^(5/2)*(-((Sqrt[c]*(15*b^3*c^3*x^3 + a*b^2*c^2*x^2*(45*c - 161*d*x) + a^2*b*c*x*(45*c^2 - 77*c*d*x + 245*d^2*x^2) + a^3*(15*c^3 - 21 *c^2*d*x + 35*c*d^2*x^2 - 105*d^3*x^3)))/(a*(a + b*x)^2)) - (105*d*(-(b*c) + a*d)^(5/2)*x^(7/2)*ArcTan[(-(d*Sqrt[x]*Sqrt[a + b*x]) + Sqrt[b]*(c + d* x))/(Sqrt[c]*Sqrt[-(b*c) + a*d])])/(a + b*x)^(5/2)))/(105*c^(9/2)*x^6)
Time = 0.51 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.90, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1261, 107, 105, 105, 105, 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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \int \frac {(a+b x)^{5/2}}{x^{9/2} (c+d x)}dx}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 107 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \int \frac {(a+b x)^{5/2}}{x^{7/2} (c+d x)}dx}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \left (\frac {(b c-a d) \int \frac {(a+b x)^{3/2}}{x^{5/2} (c+d x)}dx}{c}-\frac {2 (a+b x)^{5/2}}{5 c x^{5/2}}\right )}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {\sqrt {a+b x}}{x^{3/2} (c+d x)}dx}{c}-\frac {2 (a+b x)^{3/2}}{3 c x^{3/2}}\right )}{c}-\frac {2 (a+b x)^{5/2}}{5 c x^{5/2}}\right )}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{c}-\frac {2 \sqrt {a+b x}}{c \sqrt {x}}\right )}{c}-\frac {2 (a+b x)^{3/2}}{3 c x^{3/2}}\right )}{c}-\frac {2 (a+b x)^{5/2}}{5 c x^{5/2}}\right )}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {2 (b c-a d) \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{c}-\frac {2 \sqrt {a+b x}}{c \sqrt {x}}\right )}{c}-\frac {2 (a+b x)^{3/2}}{3 c x^{3/2}}\right )}{c}-\frac {2 (a+b x)^{5/2}}{5 c x^{5/2}}\right )}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (-\frac {d \left (\frac {(b c-a d) \left (\frac {(b c-a d) \left (\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{c^{3/2}}-\frac {2 \sqrt {a+b x}}{c \sqrt {x}}\right )}{c}-\frac {2 (a+b x)^{3/2}}{3 c x^{3/2}}\right )}{c}-\frac {2 (a+b x)^{5/2}}{5 c x^{5/2}}\right )}{c}-\frac {2 (a+b x)^{7/2}}{7 a c x^{7/2}}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
Input:
Int[(a*x + b*x^2)^(5/2)/(x^7*(c + d*x)),x]
Output:
((a*x + b*x^2)^(5/2)*((-2*(a + b*x)^(7/2))/(7*a*c*x^(7/2)) - (d*((-2*(a + b*x)^(5/2))/(5*c*x^(5/2)) + ((b*c - a*d)*((-2*(a + b*x)^(3/2))/(3*c*x^(3/2 )) + ((b*c - a*d)*((-2*Sqrt[a + b*x])/(c*Sqrt[x]) + (2*Sqrt[b*c - a*d]*Arc Tanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/c^(3/2)))/c))/c)) /c))/(x^(5/2)*(a + b*x)^(5/2))
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[(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]
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[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
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 = 0.87 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.71
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (c^{3} \left (b x +a \right )^{3}-\frac {7 d x \left (\frac {23}{3} b^{2} x^{2}+\frac {11}{3} a b x +a^{2}\right ) a \,c^{2}}{5}+\frac {7 a^{2} d^{2} x^{2} \left (7 b x +a \right ) c}{3}-7 a^{3} d^{3} x^{3}\right ) \sqrt {c \left (a d -b c \right )}\, \sqrt {x \left (b x +a \right )}+7 d a \,x^{4} \left (a d -b c \right )^{3} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )\right )}{7 \sqrt {c \left (a d -b c \right )}\, c^{4} x^{4} a}\) | \(157\) |
| risch | \(-\frac {2 \left (b x +a \right ) \left (-105 a^{3} d^{3} x^{3}+245 a^{2} b c \,d^{2} x^{3}-161 a \,b^{2} c^{2} d \,x^{3}+15 b^{3} c^{3} x^{3}+35 a^{3} c \,d^{2} x^{2}-77 a^{2} b \,c^{2} x^{2} d +45 a \,b^{2} c^{3} x^{2}-21 a^{3} c^{2} d x +45 a^{2} b \,c^{3} x +15 a^{3} c^{3}\right )}{105 a \,c^{4} \sqrt {x \left (b x +a \right )}\, x^{3}}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{4} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\) | \(308\) |
| default | \(\text {Expression too large to display}\) | \(2004\) |
Input:
int((b*x^2+a*x)^(5/2)/x^7/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/7*((c^3*(b*x+a)^3-7/5*d*x*(23/3*b^2*x^2+11/3*a*b*x+a^2)*a*c^2+7/3*a^2*d ^2*x^2*(7*b*x+a)*c-7*a^3*d^3*x^3)*(c*(a*d-b*c))^(1/2)*(x*(b*x+a))^(1/2)+7* d*a*x^4*(a*d-b*c)^3*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2)))/(c* (a*d-b*c))^(1/2)/c^4/x^4/a
Time = 0.09 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\left [\frac {105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) - 2 \, {\left (15 \, a^{3} c^{3} + {\left (15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3}\right )} x^{3} + {\left (45 \, a b^{2} c^{3} - 77 \, a^{2} b c^{2} d + 35 \, a^{3} c d^{2}\right )} x^{2} + 3 \, {\left (15 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}}{105 \, a c^{4} x^{4}}, -\frac {2 \, {\left (105 \, {\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) + {\left (15 \, a^{3} c^{3} + {\left (15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3}\right )} x^{3} + {\left (45 \, a b^{2} c^{3} - 77 \, a^{2} b c^{2} d + 35 \, a^{3} c d^{2}\right )} x^{2} + 3 \, {\left (15 \, a^{2} b c^{3} - 7 \, a^{3} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{105 \, a c^{4} x^{4}}\right ] \] Input:
integrate((b*x^2+a*x)^(5/2)/x^7/(d*x+c),x, algorithm="fricas")
Output:
[1/105*(105*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*x^4*sqrt((b*c - a*d)/c )*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x)*c*sqrt((b*c - a*d)/c))/ (d*x + c)) - 2*(15*a^3*c^3 + (15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*d ^2 - 105*a^3*d^3)*x^3 + (45*a*b^2*c^3 - 77*a^2*b*c^2*d + 35*a^3*c*d^2)*x^2 + 3*(15*a^2*b*c^3 - 7*a^3*c^2*d)*x)*sqrt(b*x^2 + a*x))/(a*c^4*x^4), -2/10 5*(105*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*x^4*sqrt(-(b*c - a*d)/c)*ar ctan(-sqrt(b*x^2 + a*x)*c*sqrt(-(b*c - a*d)/c)/((b*c - a*d)*x)) + (15*a^3* c^3 + (15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*d^2 - 105*a^3*d^3)*x^3 + (45*a*b^2*c^3 - 77*a^2*b*c^2*d + 35*a^3*c*d^2)*x^2 + 3*(15*a^2*b*c^3 - 7* a^3*c^2*d)*x)*sqrt(b*x^2 + a*x))/(a*c^4*x^4)]
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{7} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a*x)**(5/2)/x**7/(d*x+c),x)
Output:
Integral((x*(a + b*x))**(5/2)/(x**7*(c + d*x)), x)
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{{\left (d x + c\right )} x^{7}} \,d x } \] Input:
integrate((b*x^2+a*x)^(5/2)/x^7/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(5/2)/((d*x + c)*x^7), x)
Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (194) = 388\).
Time = 0.14 (sec) , antiderivative size = 600, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=-\frac {2 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{\sqrt {-b c^{2} + a c d} c^{4}} + \frac {2 \, {\left (105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} b^{3} c^{3} - 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a b^{2} c^{2} d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{2} b c d^{2} - 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{6} a^{3} d^{3} + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a b^{\frac {5}{2}} c^{3} - 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{2} b^{\frac {3}{2}} c^{2} d + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{5} a^{3} \sqrt {b} c d^{2} + 525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{2} b^{2} c^{3} - 245 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{3} b c^{2} d + 35 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{4} a^{4} c d^{2} + 525 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{3} b^{\frac {3}{2}} c^{3} - 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{4} \sqrt {b} c^{2} d + 315 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{4} b c^{3} - 21 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{5} c^{2} d + 105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{5} \sqrt {b} c^{3} + 15 \, a^{6} c^{3}\right )}}{105 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{7} c^{4}} \] Input:
integrate((b*x^2+a*x)^(5/2)/x^7/(d*x+c),x, algorithm="giac")
Output:
-2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*arctan(-((sqrt( b)*x - sqrt(b*x^2 + a*x))*d + sqrt(b)*c)/sqrt(-b*c^2 + a*c*d))/(sqrt(-b*c^ 2 + a*c*d)*c^4) + 2/105*(105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*b^3*c^3 - 3 15*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a*b^2*c^2*d + 315*(sqrt(b)*x - sqrt(b *x^2 + a*x))^6*a^2*b*c*d^2 - 105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^6*a^3*d^3 + 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a*b^(5/2)*c^3 - 315*(sqrt(b)*x - sqrt(b*x^2 + a*x))^5*a^2*b^(3/2)*c^2*d + 105*(sqrt(b)*x - sqrt(b*x^2 + a*x ))^5*a^3*sqrt(b)*c*d^2 + 525*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^2*b^2*c^3 - 245*(sqrt(b)*x - sqrt(b*x^2 + a*x))^4*a^3*b*c^2*d + 35*(sqrt(b)*x - sqr t(b*x^2 + a*x))^4*a^4*c*d^2 + 525*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^3*b^ (3/2)*c^3 - 105*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^4*sqrt(b)*c^2*d + 315* (sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^4*b*c^3 - 21*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^5*c^2*d + 105*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^5*sqrt(b)*c^3 + 15*a^6*c^3)/((sqrt(b)*x - sqrt(b*x^2 + a*x))^7*c^4)
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x^7\,\left (c+d\,x\right )} \,d x \] Input:
int((a*x + b*x^2)^(5/2)/(x^7*(c + d*x)),x)
Output:
int((a*x + b*x^2)^(5/2)/(x^7*(c + d*x)), x)
Time = 0.51 (sec) , antiderivative size = 631, normalized size of antiderivative = 2.87 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^7 (c+d x)} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{3} d^{3} x^{4}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} b c \,d^{2} x^{4}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c^{2} d \,x^{4}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{3} d^{3} x^{4}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} b c \,d^{2} x^{4}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c^{2} d \,x^{4}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{4}}{7}+\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{3} d x}{5}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c^{2} d^{2} x^{2}}{3}+2 \sqrt {x}\, \sqrt {b x +a}\, a^{3} c \,d^{3} x^{3}-\frac {6 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{4} x}{7}+\frac {22 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{3} d \,x^{2}}{15}-\frac {14 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,c^{2} d^{2} x^{3}}{3}-\frac {6 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c^{4} x^{2}}{7}+\frac {46 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c^{3} d \,x^{3}}{15}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{4} x^{3}}{7}-\frac {10 \sqrt {b}\, a^{3} c \,d^{3} x^{4}}{7}+\frac {62 \sqrt {b}\, a^{2} b \,c^{2} d^{2} x^{4}}{21}-\frac {142 \sqrt {b}\, a \,b^{2} c^{3} d \,x^{4}}{105}-\frac {2 \sqrt {b}\, b^{3} c^{4} x^{4}}{7}}{a \,c^{5} x^{4}} \] Input:
int((b*x^2+a*x)^(5/2)/x^7/(d*x+c),x)
Output:
(2*( - 105*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**3*d**3*x**4 + 210* sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sq rt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a**2*b*c*d**2*x**4 - 105*sqrt(c) *sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*s qrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b**2*c**2*d*x**4 - 105*sqrt(c)*sqrt(a *d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)* sqrt(b))/(sqrt(c)*sqrt(b)))*a**3*d**3*x**4 + 210*sqrt(c)*sqrt(a*d - b*c)*a tan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(s qrt(c)*sqrt(b)))*a**2*b*c*d**2*x**4 - 105*sqrt(c)*sqrt(a*d - b*c)*atan((sq rt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)* sqrt(b)))*a*b**2*c**2*d*x**4 - 15*sqrt(x)*sqrt(a + b*x)*a**3*c**4 + 21*sqr t(x)*sqrt(a + b*x)*a**3*c**3*d*x - 35*sqrt(x)*sqrt(a + b*x)*a**3*c**2*d**2 *x**2 + 105*sqrt(x)*sqrt(a + b*x)*a**3*c*d**3*x**3 - 45*sqrt(x)*sqrt(a + b *x)*a**2*b*c**4*x + 77*sqrt(x)*sqrt(a + b*x)*a**2*b*c**3*d*x**2 - 245*sqrt (x)*sqrt(a + b*x)*a**2*b*c**2*d**2*x**3 - 45*sqrt(x)*sqrt(a + b*x)*a*b**2* c**4*x**2 + 161*sqrt(x)*sqrt(a + b*x)*a*b**2*c**3*d*x**3 - 15*sqrt(x)*sqrt (a + b*x)*b**3*c**4*x**3 - 75*sqrt(b)*a**3*c*d**3*x**4 + 155*sqrt(b)*a**2* b*c**2*d**2*x**4 - 71*sqrt(b)*a*b**2*c**3*d*x**4 - 15*sqrt(b)*b**3*c**4*x* *4))/(105*a*c**5*x**4)