Integrand size = 24, antiderivative size = 122 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {2 (b c-a d)^2 x^3}{3 a b^2 \left (a x+b x^2\right )^{3/2}}-\frac {4 d (b c-a d) x}{b^3 \sqrt {a x+b x^2}}+\frac {d^2 \sqrt {a x+b x^2}}{b^3}+\frac {d (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{b^{7/2}} \] Output:
2/3*(-a*d+b*c)^2*x^3/a/b^2/(b*x^2+a*x)^(3/2)-4*d*(-a*d+b*c)*x/b^3/(b*x^2+a *x)^(1/2)+d^2*(b*x^2+a*x)^(1/2)/b^3+d*(-5*a*d+4*b*c)*arctanh(b^(1/2)*x/(b* x^2+a*x)^(1/2))/b^(7/2)
Time = 0.49 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.19 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {x^{5/2} \left (\frac {\sqrt {b} \sqrt {x} (a+b x) \left (15 a^3 d^2+2 b^3 c^2 x+a b^2 d x (-16 c+3 d x)+4 a^2 b d (-3 c+5 d x)\right )}{a}+6 d (4 b c-5 a d) (a+b x)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{3 b^{7/2} (x (a+b x))^{5/2}} \] Input:
Integrate[(x^3*(c + d*x)^2)/(a*x + b*x^2)^(5/2),x]
Output:
(x^(5/2)*((Sqrt[b]*Sqrt[x]*(a + b*x)*(15*a^3*d^2 + 2*b^3*c^2*x + a*b^2*d*x *(-16*c + 3*d*x) + 4*a^2*b*d*(-3*c + 5*d*x)))/a + 6*d*(4*b*c - 5*a*d)*(a + b*x)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(3*b^( 7/2)*(x*(a + b*x))^(5/2))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1242, 24}
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.
| \(\displaystyle \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1242 |
| \(\displaystyle \frac {2 x^3 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}-\frac {2 \int 0dx}{3 a^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2 x^3 \text {PolynomialRemainder}\left [(c+d x)^2,0,x\right ]}{3 a \left (a x+b x^2\right )^{3/2}}\) |
Input:
Int[(x^3*(c + d*x)^2)/(a*x + b*x^2)^(5/2),x]
Output:
(2*x^3*PolynomialRemainder[(c + d*x)^2, 0, x])/(3*a*(a*x + b*x^2)^(3/2))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(f + g*x) ^n, a*e + c*d*x, x], R = PolynomialRemainder[(f + g*x)^n, a*e + c*d*x, x]}, Simp[R*(2*c*d - b*e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^ 2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*( a + b*x + c*x^2)^(p + 1)*ExpandToSum[d*e*(p + 1)*(b^2 - 4*a*c)*Q - R*(2*c*d - b*e)*(m + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IG tQ[n, 1] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[c*d^2 - b*d*e + a*e^2, 0]
Time = 0.62 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (d \left (a d -\frac {4 b c}{5}\right ) \left (b x +a \right ) a \sqrt {x \left (b x +a \right )}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (-\frac {4 d \left (-\frac {5 d x}{3}+c \right ) a^{2} b^{\frac {3}{2}}}{5}-\frac {16 d \left (-\frac {3 d x}{16}+c \right ) x a \,b^{\frac {5}{2}}}{15}+\sqrt {b}\, a^{3} d^{2}+\frac {2 b^{\frac {7}{2}} c^{2} x}{15}\right ) x \right )}{\sqrt {x \left (b x +a \right )}\, b^{\frac {7}{2}} \left (b x +a \right ) a}\) | \(120\) |
| risch | \(\frac {d^{2} x \left (b x +a \right )}{b^{3} \sqrt {x \left (b x +a \right )}}-\frac {\frac {2 \left (-6 a^{2} d^{2}+8 a b c d -2 b^{2} c^{2}\right ) \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{b a \left (x +\frac {a}{b}\right )}+\frac {5 a \,d^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{\sqrt {b}}-4 d \sqrt {b}\, c \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )+\frac {2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a \left (\frac {2 \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{3 a \left (x +\frac {a}{b}\right )^{2}}+\frac {4 b \sqrt {\left (x +\frac {a}{b}\right )^{2} b -a \left (x +\frac {a}{b}\right )}}{3 a^{2} \left (x +\frac {a}{b}\right )}\right )}{b^{2}}}{2 b^{3}}\) | \(263\) |
| default | \(c^{2} \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )+d^{2} \left (\frac {x^{4}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {5 a \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )}{2 b}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a x}}-\frac {a \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )}{2 b}+\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}}{b}\right )}{2 b}\right )+2 c d \left (-\frac {x^{3}}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {a \left (-\frac {x}{2 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {1}{3 b \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}-\frac {a \left (-\frac {2 \left (2 b x +a \right )}{3 a^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}+\frac {16 b \left (2 b x +a \right )}{3 a^{4} \sqrt {b \,x^{2}+a x}}\right )}{2 b}\right )}{4 b}\right )}{2 b}\right )}{2 b}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a x}}-\frac {a \left (-\frac {1}{b \sqrt {b \,x^{2}+a x}}+\frac {2 b x +a}{a b \sqrt {b \,x^{2}+a x}}\right )}{2 b}+\frac {\ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{b^{\frac {3}{2}}}}{b}\right )\) | \(642\) |
Input:
int(x^3*(d*x+c)^2/(b*x^2+a*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-5/(x*(b*x+a))^(1/2)/b^(7/2)*(d*(a*d-4/5*b*c)*(b*x+a)*a*(x*(b*x+a))^(1/2)* arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-(-4/5*d*(-5/3*d*x+c)*a^2*b^(3/2)-16/1 5*d*(-3/16*d*x+c)*x*a*b^(5/2)+b^(1/2)*a^3*d^2+2/15*b^(7/2)*c^2*x)*x)/(b*x+ a)/a
Time = 0.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.27 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (4 \, a^{3} b c d - 5 \, a^{4} d^{2} + {\left (4 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (4 \, a^{2} b^{2} c d - 5 \, a^{3} b d^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (3 \, a b^{3} d^{2} x^{2} - 12 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 8 \, a b^{3} c d + 10 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{6 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac {3 \, {\left (4 \, a^{3} b c d - 5 \, a^{4} d^{2} + {\left (4 \, a b^{3} c d - 5 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (4 \, a^{2} b^{2} c d - 5 \, a^{3} b d^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (3 \, a b^{3} d^{2} x^{2} - 12 \, a^{2} b^{2} c d + 15 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 8 \, a b^{3} c d + 10 \, a^{2} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{3 \, {\left (a b^{6} x^{2} + 2 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \] Input:
integrate(x^3*(d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="fricas")
Output:
[-1/6*(3*(4*a^3*b*c*d - 5*a^4*d^2 + (4*a*b^3*c*d - 5*a^2*b^2*d^2)*x^2 + 2* (4*a^2*b^2*c*d - 5*a^3*b*d^2)*x)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a* x)*sqrt(b)) - 2*(3*a*b^3*d^2*x^2 - 12*a^2*b^2*c*d + 15*a^3*b*d^2 + 2*(b^4* c^2 - 8*a*b^3*c*d + 10*a^2*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/(a*b^6*x^2 + 2*a ^2*b^5*x + a^3*b^4), -1/3*(3*(4*a^3*b*c*d - 5*a^4*d^2 + (4*a*b^3*c*d - 5*a ^2*b^2*d^2)*x^2 + 2*(4*a^2*b^2*c*d - 5*a^3*b*d^2)*x)*sqrt(-b)*arctan(sqrt( b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (3*a*b^3*d^2*x^2 - 12*a^2*b^2*c*d + 15* a^3*b*d^2 + 2*(b^4*c^2 - 8*a*b^3*c*d + 10*a^2*b^2*d^2)*x)*sqrt(b*x^2 + a*x ))/(a*b^6*x^2 + 2*a^2*b^5*x + a^3*b^4)]
\[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (c + d x\right )^{2}}{\left (x \left (a + b x\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**3*(d*x+c)**2/(b*x**2+a*x)**(5/2),x)
Output:
Integral(x**3*(c + d*x)**2/(x*(a + b*x))**(5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (108) = 216\).
Time = 0.05 (sec) , antiderivative size = 410, normalized size of antiderivative = 3.36 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=-\frac {2}{3} \, c d x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )} + \frac {5 \, a d^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{6 \, b} + \frac {d^{2} x^{4}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {c^{2} x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {a c^{2} x}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {2 \, c^{2} x}{3 \, \sqrt {b x^{2} + a x} a b} - \frac {8 \, c d x}{3 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {10 \, a d^{2} x}{3 \, \sqrt {b x^{2} + a x} b^{3}} + \frac {2 \, c d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{b^{\frac {5}{2}}} - \frac {5 \, a d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{2 \, b^{\frac {7}{2}}} + \frac {c^{2}}{3 \, \sqrt {b x^{2} + a x} b^{2}} - \frac {4 \, \sqrt {b x^{2} + a x} c d}{3 \, a b^{2}} + \frac {5 \, \sqrt {b x^{2} + a x} d^{2}}{3 \, b^{3}} \] Input:
integrate(x^3*(d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="maxima")
Output:
-2/3*c*d*x*(3*x^2/((b*x^2 + a*x)^(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) - 2*x/(sqrt(b*x^2 + a*x)*a*b) - 1/(sqrt(b*x^2 + a*x)*b^2)) + 5/6*a*d^2*x*( 3*x^2/((b*x^2 + a*x)^(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) - 2*x/(sqrt( b*x^2 + a*x)*a*b) - 1/(sqrt(b*x^2 + a*x)*b^2))/b + d^2*x^4/((b*x^2 + a*x)^ (3/2)*b) - c^2*x^2/((b*x^2 + a*x)^(3/2)*b) - 1/3*a*c^2*x/((b*x^2 + a*x)^(3 /2)*b^2) + 2/3*c^2*x/(sqrt(b*x^2 + a*x)*a*b) - 8/3*c*d*x/(sqrt(b*x^2 + a*x )*b^2) + 10/3*a*d^2*x/(sqrt(b*x^2 + a*x)*b^3) + 2*c*d*log(2*b*x + a + 2*sq rt(b*x^2 + a*x)*sqrt(b))/b^(5/2) - 5/2*a*d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 1/3*c^2/(sqrt(b*x^2 + a*x)*b^2) - 4/3*sqrt(b*x^2 + a*x)*c*d/(a*b^2) + 5/3*sqrt(b*x^2 + a*x)*d^2/b^3
Leaf count of result is larger than twice the leaf count of optimal. 298 vs. \(2 (108) = 216\).
Time = 0.21 (sec) , antiderivative size = 298, normalized size of antiderivative = 2.44 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {\sqrt {b x^{2} + a x} d^{2}}{b^{3}} - \frac {{\left (4 \, b c d - 5 \, a d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{2 \, b^{\frac {7}{2}}} + \frac {2 \, {\left (3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b^{3} c^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a b^{2} c d + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} b d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a b^{\frac {5}{2}} c^{2} - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{2} b^{\frac {3}{2}} c d + 15 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} \sqrt {b} d^{2} + a^{2} b^{2} c^{2} - 8 \, a^{3} b c d + 7 \, a^{4} d^{2}\right )}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a\right )}^{3} b^{\frac {7}{2}}} \] Input:
integrate(x^3*(d*x+c)^2/(b*x^2+a*x)^(5/2),x, algorithm="giac")
Output:
sqrt(b*x^2 + a*x)*d^2/b^3 - 1/2*(4*b*c*d - 5*a*d^2)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(7/2) + 2/3*(3*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*b^3*c^2 - 12*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a*b^2*c*d + 9*(s qrt(b)*x - sqrt(b*x^2 + a*x))^2*a^2*b*d^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a* x))*a*b^(5/2)*c^2 - 18*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^2*b^(3/2)*c*d + 1 5*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^3*sqrt(b)*d^2 + a^2*b^2*c^2 - 8*a^3*b* c*d + 7*a^4*d^2)/(((sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a)^3*b^(7/2))
Timed out. \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\left (c+d\,x\right )}^2}{{\left (b\,x^2+a\,x\right )}^{5/2}} \,d x \] Input:
int((x^3*(c + d*x)^2)/(a*x + b*x^2)^(5/2),x)
Output:
int((x^3*(c + d*x)^2)/(a*x + b*x^2)^(5/2), x)
Time = 0.19 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.49 \[ \int \frac {x^3 (c+d x)^2}{\left (a x+b x^2\right )^{5/2}} \, dx=\frac {-30 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{2}+24 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c d -30 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b \,d^{2} x +24 \sqrt {b}\, \sqrt {b x +a}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{2} c d x -5 \sqrt {b}\, \sqrt {b x +a}\, a^{3} d^{2}-5 \sqrt {b}\, \sqrt {b x +a}\, a^{2} b \,d^{2} x +4 \sqrt {b}\, \sqrt {b x +a}\, a \,b^{2} c^{2}+4 \sqrt {b}\, \sqrt {b x +a}\, b^{3} c^{2} x +30 \sqrt {x}\, a^{3} b \,d^{2}-24 \sqrt {x}\, a^{2} b^{2} c d +40 \sqrt {x}\, a^{2} b^{2} d^{2} x -32 \sqrt {x}\, a \,b^{3} c d x +6 \sqrt {x}\, a \,b^{3} d^{2} x^{2}+4 \sqrt {x}\, b^{4} c^{2} x}{6 \sqrt {b x +a}\, a \,b^{4} \left (b x +a \right )} \] Input:
int(x^3*(d*x+c)^2/(b*x^2+a*x)^(5/2),x)
Output:
( - 30*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a) )*a**3*d**2 + 24*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b ))/sqrt(a))*a**2*b*c*d - 30*sqrt(b)*sqrt(a + b*x)*log((sqrt(a + b*x) + sqr t(x)*sqrt(b))/sqrt(a))*a**2*b*d**2*x + 24*sqrt(b)*sqrt(a + b*x)*log((sqrt( a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b**2*c*d*x - 5*sqrt(b)*sqrt(a + b*x )*a**3*d**2 - 5*sqrt(b)*sqrt(a + b*x)*a**2*b*d**2*x + 4*sqrt(b)*sqrt(a + b *x)*a*b**2*c**2 + 4*sqrt(b)*sqrt(a + b*x)*b**3*c**2*x + 30*sqrt(x)*a**3*b* d**2 - 24*sqrt(x)*a**2*b**2*c*d + 40*sqrt(x)*a**2*b**2*d**2*x - 32*sqrt(x) *a*b**3*c*d*x + 6*sqrt(x)*a*b**3*d**2*x**2 + 4*sqrt(x)*b**4*c**2*x)/(6*sqr t(a + b*x)*a*b**4*(a + b*x))