Integrand size = 24, antiderivative size = 142 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=-\frac {2 a (2 b c-a d) \sqrt {a x+b x^2}}{c^2 x}-\frac {2 a \left (a x+b x^2\right )^{3/2}}{3 c x^3}+\frac {2 b^{5/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{d}-\frac {2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{c^{5/2} d} \] Output:
-2*a*(-a*d+2*b*c)*(b*x^2+a*x)^(1/2)/c^2/x-2/3*a*(b*x^2+a*x)^(3/2)/c/x^3+2* b^(5/2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/d-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))/c^(5/2)/d
Time = 0.58 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=-\frac {2 \sqrt {x (a+b x)} \left (3 (-b c+a d)^{5/2} x^{3/2} \arctan \left (\frac {-d \sqrt {x} \sqrt {a+b x}+\sqrt {b} (c+d x)}{\sqrt {c} \sqrt {-b c+a d}}\right )+\sqrt {c} \left (a d \sqrt {a+b x} (7 b c x+a (c-3 d x))+3 b^{5/2} c^2 x^{3/2} \log \left (-\sqrt {b} \sqrt {x}+\sqrt {a+b x}\right )\right )\right )}{3 c^{5/2} d x^2 \sqrt {a+b x}} \] Input:
Integrate[(a*x + b*x^2)^(5/2)/(x^5*(c + d*x)),x]
Output:
(-2*Sqrt[x*(a + b*x)]*(3*(-(b*c) + a*d)^(5/2)*x^(3/2)*ArcTan[(-(d*Sqrt[x]* Sqrt[a + b*x]) + Sqrt[b]*(c + d*x))/(Sqrt[c]*Sqrt[-(b*c) + a*d])] + Sqrt[c ]*(a*d*Sqrt[a + b*x]*(7*b*c*x + a*(c - 3*d*x)) + 3*b^(5/2)*c^2*x^(3/2)*Log [-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])))/(3*c^(5/2)*d*x^2*Sqrt[a + b*x])
Leaf count is larger than twice the leaf count of optimal. \(904\) vs. \(2(142)=284\).
Time = 2.13 (sec) , antiderivative size = 904, normalized size of antiderivative = 6.37, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1260, 2009}
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.
| \(\displaystyle \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx\) |
| \(\Big \downarrow \) 1260 |
| \(\displaystyle \int \left (-\frac {d^5 \left (a x+b x^2\right )^{5/2}}{c^5 (c+d x)}+\frac {d^4 \left (a x+b x^2\right )^{5/2}}{c^5 x}-\frac {d^3 \left (a x+b x^2\right )^{5/2}}{c^4 x^2}+\frac {d^2 \left (a x+b x^2\right )^{5/2}}{c^3 x^3}-\frac {d \left (a x+b x^2\right )^{5/2}}{c^2 x^4}+\frac {\left (a x+b x^2\right )^{5/2}}{c x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 d^4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right ) a^5}{128 b^{5/2} c^5}+\frac {5 d^3 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right ) a^4}{64 b^{3/2} c^4}+\frac {5 d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right ) a^3}{8 \sqrt {b} c^3}-\frac {3 d^4 (a+2 b x) \sqrt {b x^2+a x} a^3}{128 b^2 c^5}-\frac {15 \sqrt {b} d \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right ) a^2}{4 c^2}-\frac {5 d^3 (a+2 b x) \sqrt {b x^2+a x} a^2}{64 b c^4}+\frac {2 d \sqrt {b x^2+a x} a^2}{c^2 x}-\frac {5 d^3 \left (b x^2+a x\right )^{3/2} a}{24 c^4}+\frac {d^4 (a+2 b x) \left (b x^2+a x\right )^{3/2} a}{16 b c^5}+\frac {5 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right ) a}{c}-\frac {9 b d \sqrt {b x^2+a x} a}{4 c^2}-\frac {5 d^2 (a+2 b x) \sqrt {b x^2+a x} a}{8 c^3}-\frac {10 b \sqrt {b x^2+a x} a}{3 c x}-\frac {d^3 \left (b x^2+a x\right )^{5/2}}{4 c^4 x}+\frac {2 d^2 \left (b x^2+a x\right )^{5/2}}{c^3 x^2}-\frac {2 \left (b x^2+a x\right )^{5/2}}{3 c x^4}-\frac {5 b d^2 \left (b x^2+a x\right )^{3/2}}{3 c^3}-\frac {d^2 \left (16 b^2 c^2-22 a b d c+3 a^2 d^2-6 b d (2 b c-a d) x\right ) \left (b x^2+a x\right )^{3/2}}{48 b c^5}+\frac {(2 b c-a d) \left (128 b^4 c^4-256 a b^3 d c^3+112 a^2 b^2 d^2 c^2+16 a^3 b d^3 c+3 a^4 d^4\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+a x}}\right )}{128 b^{5/2} c^5 d}-\frac {(b c-a d)^{5/2} \text {arctanh}\left (\frac {a c+(2 b c-a d) x}{2 \sqrt {c} \sqrt {b c-a d} \sqrt {b x^2+a x}}\right )}{c^{5/2} d}-\frac {b^2 d x \sqrt {b x^2+a x}}{2 c^2}-\frac {\left (128 b^4 c^4-288 a b^3 d c^3+176 a^2 b^2 d^2 c^2-10 a^3 b d^3 c-3 a^4 d^4-2 b d (2 b c-a d) \left (16 b^2 c^2-16 a b d c-3 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}}{128 b^2 c^5}+\frac {5 b^2 \sqrt {b x^2+a x}}{3 c}\) |
Input:
Int[(a*x + b*x^2)^(5/2)/(x^5*(c + d*x)),x]
Output:
(5*b^2*Sqrt[a*x + b*x^2])/(3*c) - (9*a*b*d*Sqrt[a*x + b*x^2])/(4*c^2) - (1 0*a*b*Sqrt[a*x + b*x^2])/(3*c*x) + (2*a^2*d*Sqrt[a*x + b*x^2])/(c^2*x) - ( b^2*d*x*Sqrt[a*x + b*x^2])/(2*c^2) - (5*a*d^2*(a + 2*b*x)*Sqrt[a*x + b*x^2 ])/(8*c^3) - (5*a^2*d^3*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(64*b*c^4) - (3*a^3 *d^4*(a + 2*b*x)*Sqrt[a*x + b*x^2])/(128*b^2*c^5) - ((128*b^4*c^4 - 288*a* b^3*c^3*d + 176*a^2*b^2*c^2*d^2 - 10*a^3*b*c*d^3 - 3*a^4*d^4 - 2*b*d*(2*b* c - a*d)*(16*b^2*c^2 - 16*a*b*c*d - 3*a^2*d^2)*x)*Sqrt[a*x + b*x^2])/(128* b^2*c^5) - (5*b*d^2*(a*x + b*x^2)^(3/2))/(3*c^3) - (5*a*d^3*(a*x + b*x^2)^ (3/2))/(24*c^4) + (a*d^4*(a + 2*b*x)*(a*x + b*x^2)^(3/2))/(16*b*c^5) - (d^ 2*(16*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2 - 6*b*d*(2*b*c - a*d)*x)*(a*x + b*x ^2)^(3/2))/(48*b*c^5) - (2*(a*x + b*x^2)^(5/2))/(3*c*x^4) + (2*d^2*(a*x + b*x^2)^(5/2))/(c^3*x^2) - (d^3*(a*x + b*x^2)^(5/2))/(4*c^4*x) + (5*a*b^(3/ 2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/c - (15*a^2*Sqrt[b]*d*ArcTanh[( Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(4*c^2) + (5*a^3*d^2*ArcTanh[(Sqrt[b]*x)/Sq rt[a*x + b*x^2]])/(8*Sqrt[b]*c^3) + (5*a^4*d^3*ArcTanh[(Sqrt[b]*x)/Sqrt[a* x + b*x^2]])/(64*b^(3/2)*c^4) + (3*a^5*d^4*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(128*b^(5/2)*c^5) + ((2*b*c - a*d)*(128*b^4*c^4 - 256*a*b^3*c^3*d + 112*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + 3*a^4*d^4)*ArcTanh[(Sqrt[b]*x)/S qrt[a*x + b*x^2]])/(128*b^(5/2)*c^5*d) - ((b*c - a*d)^(5/2)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/(c^(5...
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p , (d + e*x)^m*(f + g*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ [c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + n + 2*p + 1, 0] && ILtQ[m, 0] && ILtQ [n, 0]
Time = 0.73 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.95
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (3 x^{2} \left (a d -b c \right )^{3} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\left (-3 b^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right ) c^{2} x^{2}+d \sqrt {x \left (b x +a \right )}\, \left (\left (7 b x +a \right ) c -3 a d x \right ) a \right ) \sqrt {c \left (a d -b c \right )}\right )}{3 \sqrt {c \left (a d -b c \right )}\, c^{2} x^{2} d}\) | \(135\) |
| risch | \(-\frac {2 a \left (b x +a \right ) \left (-3 a d x +7 c b x +a c \right )}{3 c^{2} \sqrt {x \left (b x +a \right )}\, x}+\frac {\frac {b^{\frac {5}{2}} c^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d}-\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{c^{2}}\) | \(245\) |
| default | \(\text {Expression too large to display}\) | \(1738\) |
Input:
int((b*x^2+a*x)^(5/2)/x^5/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2/3*(3*x^2*(a*d-b*c)^3*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+ (-3*b^(5/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))*c^2*x^2+d*(x*(b*x+a))^(1/ 2)*((7*b*x+a)*c-3*a*d*x)*a)*(c*(a*d-b*c))^(1/2))/(c*(a*d-b*c))^(1/2)/c^2/x ^2/d
Time = 0.20 (sec) , antiderivative size = 698, normalized size of antiderivative = 4.92 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\left [\frac {3 \, b^{\frac {5}{2}} c^{2} x^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) - 2 \, {\left (a^{2} c d + {\left (7 \, a b c d - 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, c^{2} d x^{2}}, \frac {3 \, b^{\frac {5}{2}} c^{2} x^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) - 2 \, {\left (a^{2} c d + {\left (7 \, a b c d - 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, c^{2} d x^{2}}, -\frac {6 \, \sqrt {-b} b^{2} c^{2} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} c \sqrt {\frac {b c - a d}{c}}}{d x + c}\right ) + 2 \, {\left (a^{2} c d + {\left (7 \, a b c d - 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, c^{2} d x^{2}}, -\frac {2 \, {\left (3 \, \sqrt {-b} b^{2} c^{2} x^{2} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \sqrt {-\frac {b c - a d}{c}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} c \sqrt {-\frac {b c - a d}{c}}}{{\left (b c - a d\right )} x}\right ) + {\left (a^{2} c d + {\left (7 \, a b c d - 3 \, a^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}\right )}}{3 \, c^{2} d x^{2}}\right ] \] Input:
integrate((b*x^2+a*x)^(5/2)/x^5/(d*x+c),x, algorithm="fricas")
Output:
[1/3*(3*b^(5/2)*c^2*x^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 3*( b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*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*(a^2*c *d + (7*a*b*c*d - 3*a^2*d^2)*x)*sqrt(b*x^2 + a*x))/(c^2*d*x^2), 1/3*(3*b^( 5/2)*c^2*x^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 6*(b^2*c^2 - 2 *a*b*c*d + a^2*d^2)*x^2*sqrt(-(b*c - a*d)/c)*arctan(-sqrt(b*x^2 + a*x)*c*s qrt(-(b*c - a*d)/c)/((b*c - a*d)*x)) - 2*(a^2*c*d + (7*a*b*c*d - 3*a^2*d^2 )*x)*sqrt(b*x^2 + a*x))/(c^2*d*x^2), -1/3*(6*sqrt(-b)*b^2*c^2*x^2*arctan(s qrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x ^2*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*(a^2*c*d + (7*a*b*c*d - 3*a^2*d^2)*x)* sqrt(b*x^2 + a*x))/(c^2*d*x^2), -2/3*(3*sqrt(-b)*b^2*c^2*x^2*arctan(sqrt(b *x^2 + a*x)*sqrt(-b)/(b*x + a)) + 3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*x^2*sq rt(-(b*c - a*d)/c)*arctan(-sqrt(b*x^2 + a*x)*c*sqrt(-(b*c - a*d)/c)/((b*c - a*d)*x)) + (a^2*c*d + (7*a*b*c*d - 3*a^2*d^2)*x)*sqrt(b*x^2 + a*x))/(c^2 *d*x^2)]
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\int \frac {\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}{x^{5} \left (c + d x\right )}\, dx \] Input:
integrate((b*x**2+a*x)**(5/2)/x**5/(d*x+c),x)
Output:
Integral((x*(a + b*x))**(5/2)/(x**5*(c + d*x)), x)
\[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\int { \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}}}{{\left (d x + c\right )} x^{5}} \,d x } \] Input:
integrate((b*x^2+a*x)^(5/2)/x^5/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^2 + a*x)^(5/2)/((d*x + c)*x^5), x)
Exception generated. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((b*x^2+a*x)^(5/2)/x^5/(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. \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\,x\right )}^{5/2}}{x^5\,\left (c+d\,x\right )} \,d x \] Input:
int((a*x + b*x^2)^(5/2)/(x^5*(c + d*x)),x)
Output:
int((a*x + b*x^2)^(5/2)/(x^5*(c + d*x)), x)
Time = 0.29 (sec) , antiderivative size = 471, normalized size of antiderivative = 3.32 \[ \int \frac {\left (a x+b x^2\right )^{5/2}}{x^5 (c+d x)} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b c d \,x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{2} c^{2} x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a^{2} d^{2} x^{2}+4 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a b c d \,x^{2}-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{2} c^{2} x^{2}-\frac {2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c^{2} d}{3}+2 \sqrt {x}\, \sqrt {b x +a}\, a^{2} c \,d^{2} x -\frac {14 \sqrt {x}\, \sqrt {b x +a}\, a b \,c^{2} d x}{3}+2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{2} c^{3} x^{2}-\frac {2 \sqrt {b}\, a^{2} c \,d^{2} x^{2}}{3}+\frac {2 \sqrt {b}\, a b \,c^{2} d \,x^{2}}{3}}{c^{3} d \,x^{2}} \] Input:
int((b*x^2+a*x)^(5/2)/x^5/(d*x+c),x)
Output:
(2*( - 3*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*d**2*x**2 + 6*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*b*c*d*x**2 - 3*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)))*b**2*c**2*x**2 - 3*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**2*d**2*x**2 + 6*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*b *c*d*x**2 - 3*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)))*b**2*c**2*x**2 - s qrt(x)*sqrt(a + b*x)*a**2*c**2*d + 3*sqrt(x)*sqrt(a + b*x)*a**2*c*d**2*x - 7*sqrt(x)*sqrt(a + b*x)*a*b*c**2*d*x + 3*sqrt(b)*log((sqrt(a + b*x) + sqr t(x)*sqrt(b))/sqrt(a))*b**2*c**3*x**2 - sqrt(b)*a**2*c*d**2*x**2 + sqrt(b) *a*b*c**2*d*x**2))/(3*c**3*d*x**2)