Integrand size = 24, antiderivative size = 219 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\frac {(4 b c-5 a d) (2 b c-a d) \sqrt {a x+b x^2}}{8 d^3}-\frac {(6 b c-5 a d) \left (a x+b x^2\right )^{3/2}}{12 d^2 x}+\frac {\left (a x+b x^2\right )^{5/2}}{3 d x^2}-\frac {\left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{8 \sqrt {b} d^4}+\frac {2 \sqrt {c} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^4} \] Output:
1/8*(-5*a*d+4*b*c)*(-a*d+2*b*c)*(b*x^2+a*x)^(1/2)/d^3-1/12*(-5*a*d+6*b*c)* (b*x^2+a*x)^(3/2)/d^2/x+1/3*(b*x^2+a*x)^(5/2)/d/x^2-1/8*(-5*a^3*d^3+30*a^2 *b*c*d^2-40*a*b^2*c^2*d+16*b^3*c^3)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b ^(1/2)/d^4+2*c^(1/2)*(-a*d+b*c)^(5/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/( b*x^2+a*x)^(1/2))/d^4
Result contains complex when optimal does not.
Time = 3.18 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.28 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\frac {(x (a+b x))^{5/2} \left (b \sqrt {c} d \sqrt {x} \sqrt {a+b x} \left (33 a^2 d^2+2 a b d (-27 c+13 d x)+4 b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )-48 (b c-a d)^2 \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )-48 (b c-a d)^2 \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (-\sqrt {a}+\sqrt {a+b x}\right )}\right )+6 \sqrt {b} \sqrt {c} \left (16 b^3 c^3-40 a b^2 c^2 d+30 a^2 b c d^2-5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )\right )}{24 b \sqrt {c} d^4 x^{5/2} (a+b x)^{5/2}} \] Input:
Integrate[(a*x + b*x^2)^(5/2)/(x^2*(c + d*x)),x]
Output:
((x*(a + b*x))^(5/2)*(b*Sqrt[c]*d*Sqrt[x]*Sqrt[a + b*x]*(33*a^2*d^2 + 2*a* b*d*(-27*c + 13*d*x) + 4*b^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2)) - 48*(b*c - a* d)^2*(b*c - a*d - I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[-(b*c) + 2*a*d - (2*I )*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c]*(-Sqrt[a] + Sqrt[a + b*x]))] - 48*(b*c - a*d)^2*(b*c - a*d + I*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]) *Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt [-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(Sqrt[c] *(-Sqrt[a] + Sqrt[a + b*x]))] + 6*Sqrt[b]*Sqrt[c]*(16*b^3*c^3 - 40*a*b^2*c ^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sq rt[a + b*x])]))/(24*b*Sqrt[c]*d^4*x^(5/2)*(a + b*x)^(5/2))
Time = 0.82 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1261, 112, 27, 171, 27, 171, 27, 175, 65, 104, 219, 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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1261 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \int \frac {\sqrt {x} (a+b x)^{5/2}}{c+d x}dx}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\int \frac {(a+b x)^{3/2} (a c+(6 b c-5 a d) x)}{2 \sqrt {x} (c+d x)}dx}{3 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\int \frac {(a+b x)^{3/2} (a c+(6 b c-5 a d) x)}{\sqrt {x} (c+d x)}dx}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\int -\frac {3 \sqrt {a+b x} (a c (2 b c-3 a d)+(4 b c-5 a d) (2 b c-a d) x)}{2 \sqrt {x} (c+d x)}dx}{2 d}+\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \int \frac {\sqrt {a+b x} (a c (2 b c-3 a d)+(4 b c-5 a d) (2 b c-a d) x)}{\sqrt {x} (c+d x)}dx}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\int -\frac {a c \left (8 b^2 c^2-18 a b d c+11 a^2 d^2\right )+\left (16 b^3 c^3-40 a b^2 d c^2+30 a^2 b d^2 c-5 a^3 d^3\right ) x}{2 \sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}+\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\int \frac {a c \left (8 b^2 c^2-18 a b d c+11 a^2 d^2\right )+\left (16 b^3 c^3-40 a b^2 d c^2+30 a^2 b d^2 c-5 a^3 d^3\right ) x}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\frac {\left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {16 c (b c-a d)^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 65 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\frac {2 \left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {16 c (b c-a d)^3 \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\frac {2 \left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {32 c (b c-a d)^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right )}{\sqrt {b} d}-\frac {32 c (b c-a d)^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (a x+b x^2\right )^{5/2} \left (\frac {\sqrt {x} (a+b x)^{5/2}}{3 d}-\frac {\frac {\sqrt {x} (a+b x)^{3/2} (6 b c-5 a d)}{2 d}-\frac {3 \left (\frac {\sqrt {x} \sqrt {a+b x} (4 b c-5 a d) (2 b c-a d)}{d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (-5 a^3 d^3+30 a^2 b c d^2-40 a b^2 c^2 d+16 b^3 c^3\right )}{\sqrt {b} d}-\frac {32 \sqrt {c} (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{d}}{2 d}\right )}{4 d}}{6 d}\right )}{x^{5/2} (a+b x)^{5/2}}\) |
Input:
Int[(a*x + b*x^2)^(5/2)/(x^2*(c + d*x)),x]
Output:
((a*x + b*x^2)^(5/2)*((Sqrt[x]*(a + b*x)^(5/2))/(3*d) - (((6*b*c - 5*a*d)* Sqrt[x]*(a + b*x)^(3/2))/(2*d) - (3*(((4*b*c - 5*a*d)*(2*b*c - a*d)*Sqrt[x ]*Sqrt[a + b*x])/d - ((2*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5 *a^3*d^3)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(Sqrt[b]*d) - (32*Sqrt [c]*(b*c - a*d)^(5/2)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/d)/(2*d)))/(4*d))/(6*d)))/(x^(5/2)*(a + b*x)^(5/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2 Sub st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d }, x] && !GtQ[c, 0]
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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(b*g - a*h)/b Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 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]
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 = 0.67 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {-\frac {d \sqrt {x \left (b x +a \right )}\, \left (8 b^{2} d^{2} x^{2}+26 a b \,d^{2} x -12 b^{2} c x d +33 a^{2} d^{2}-54 a b c d +24 b^{2} c^{2}\right )}{24}-\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{8 \sqrt {b}}-\frac {2 \left (a d -b c \right )^{3} c \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{d^{4}}\) | \(180\) |
| risch | \(\frac {\left (8 b^{2} d^{2} x^{2}+26 a b \,d^{2} x -12 b^{2} c x d +33 a^{2} d^{2}-54 a b c d +24 b^{2} c^{2}\right ) x \left (b x +a \right )}{24 d^{3} \sqrt {x \left (b x +a \right )}}+\frac {\frac {\left (5 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {16 c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{16 d^{3}}\) | \(314\) |
| default | \(\text {Expression too large to display}\) | \(1170\) |
Input:
int((b*x^2+a*x)^(5/2)/x^2/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/d^4*(-1/24*d*(x*(b*x+a))^(1/2)*(8*b^2*d^2*x^2+26*a*b*d^2*x-12*b^2*c*d*x +33*a^2*d^2-54*a*b*c*d+24*b^2*c^2)-1/8*(5*a^3*d^3-30*a^2*b*c*d^2+40*a*b^2* c^2*d-16*b^3*c^3)/b^(1/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-2*(a*d-b*c) ^3*c/(c*(a*d-b*c))^(1/2)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2)) )
Time = 0.29 (sec) , antiderivative size = 932, normalized size of antiderivative = 4.26 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a*x)^(5/2)/x^2/(d*x+c),x, algorithm="fricas")
Output:
[-1/48*(3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*sqrt( b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 48*(b^3*c^2 - 2*a*b^2*c* d + a^2*b*d^2)*sqrt(b*c^2 - a*c*d)*log((a*c + (2*b*c - a*d)*x + 2*sqrt(b*c ^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - 2*(8*b^3*d^3*x^2 + 24*b^3*c^2* d - 54*a*b^2*c*d^2 + 33*a^2*b*d^3 - 2*(6*b^3*c*d^2 - 13*a*b^2*d^3)*x)*sqrt (b*x^2 + a*x))/(b*d^4), -1/48*(96*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt (-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)*sqrt(b*x^2 + a*x)/(b*c*x + a* c)) + 3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c*d^2 - 5*a^3*d^3)*sqrt(b) *log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(8*b^3*d^3*x^2 + 24*b^3* c^2*d - 54*a*b^2*c*d^2 + 33*a^2*b*d^3 - 2*(6*b^3*c*d^2 - 13*a*b^2*d^3)*x)* sqrt(b*x^2 + a*x))/(b*d^4), 1/24*(3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2* b*c*d^2 - 5*a^3*d^3)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + 24*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*sqrt(b*c^2 - a*c*d)*log((a*c + ( 2*b*c - a*d)*x + 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) + (8* b^3*d^3*x^2 + 24*b^3*c^2*d - 54*a*b^2*c*d^2 + 33*a^2*b*d^3 - 2*(6*b^3*c*d^ 2 - 13*a*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/(b*d^4), -1/24*(48*(b^3*c^2 - 2*a* b^2*c*d + a^2*b*d^2)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)*sqrt (b*x^2 + a*x)/(b*c*x + a*c)) - 3*(16*b^3*c^3 - 40*a*b^2*c^2*d + 30*a^2*b*c *d^2 - 5*a^3*d^3)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (8*b^3*d^3*x^2 + 24*b^3*c^2*d - 54*a*b^2*c*d^2 + 33*a^2*b*d^3 - 2*(6*b^...
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{2} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a*x)**(5/2)/x**2/(d*x+c),x)
Output:
Integral((x*(a + b*x))**(5/2)/(x**2*(c + d*x)), x)
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{{\left (d x + c\right )} x^{2}} \,d x } \] Input:
integrate((b*x^2+a*x)^(5/2)/x^2/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(5/2)/((d*x + c)*x^2), x)
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a*x)^(5/2)/x^2/(d*x+c),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x^2\,\left (c+d\,x\right )} \,d x \] Input:
int((a*x + b*x^2)^(5/2)/(x^2*(c + d*x)),x)
Output:
int((a*x + b*x^2)^(5/2)/(x^2*(c + d*x)), x)
Time = 0.29 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.60 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^2 (c+d x)} \, dx=\frac {48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} b \,d^{2}-96 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}+48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} b \,d^{2}-96 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}+33 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{3}-54 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c \,d^{2}+26 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} d^{3} x +24 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{2} d -12 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c \,d^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, b^{3} d^{3} x^{2}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{3}-90 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c \,d^{2}+120 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{2} c^{2} d -48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{3} c^{3}}{24 b \,d^{4}} \] Input:
int((b*x^2+a*x)^(5/2)/x^2/(d*x+c),x)
Output:
(48*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**2*b*d**2 - 96*sqrt(c)*sqr t(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*b**2*c*d + 48*sqrt(c)*sqrt(a*d - b*c)*ata n((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqr t(c)*sqrt(b)))*b**3*c**2 + 48*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* *2*b*d**2 - 96*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqr t(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b**2*c*d + 48*s qrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqr t(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**3*c**2 + 33*sqrt(x)*sqrt(a + b *x)*a**2*b*d**3 - 54*sqrt(x)*sqrt(a + b*x)*a*b**2*c*d**2 + 26*sqrt(x)*sqrt (a + b*x)*a*b**2*d**3*x + 24*sqrt(x)*sqrt(a + b*x)*b**3*c**2*d - 12*sqrt(x )*sqrt(a + b*x)*b**3*c*d**2*x + 8*sqrt(x)*sqrt(a + b*x)*b**3*d**3*x**2 + 1 5*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**3*d**3 - 90*sq rt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*b*c*d**2 + 120*s qrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b**2*c**2*d - 48*s qrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b**3*c**3)/(24*b*d** 4)