Integrand size = 24, antiderivative size = 208 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx=\frac {2 (b c-a d)^2 (d e-c f)^2 \sqrt {e+f x}}{d^5}+\frac {2 (b c-a d)^2 (d e-c f) (e+f x)^{3/2}}{3 d^4}+\frac {2 (b c-a d)^2 (e+f x)^{5/2}}{5 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{7/2}}{7 d^2 f^2}+\frac {2 b^2 (e+f x)^{9/2}}{9 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}} \] Output:
2*(-a*d+b*c)^2*(-c*f+d*e)^2*(f*x+e)^(1/2)/d^5+2/3*(-a*d+b*c)^2*(-c*f+d*e)* (f*x+e)^(3/2)/d^4+2/5*(-a*d+b*c)^2*(f*x+e)^(5/2)/d^3-2/7*b*(-2*a*d*f+b*c*f +b*d*e)*(f*x+e)^(7/2)/d^2/f^2+2/9*b^2*(f*x+e)^(9/2)/d/f^2-2*(-a*d+b*c)^2*( -c*f+d*e)^(5/2)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(11/2)
Time = 0.37 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (21 a^2 d^2 f^2 \left (15 c^2 f^2-5 c d f (7 e+f x)+d^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+6 a b d f \left (-105 c^3 f^3+15 d^3 (e+f x)^3+35 c^2 d f^2 (7 e+f x)-7 c d^2 f \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )+b^2 \left (315 c^4 f^4-45 c d^3 f (e+f x)^3-5 d^4 (2 e-7 f x) (e+f x)^3-105 c^3 d f^3 (7 e+f x)+21 c^2 d^2 f^2 \left (23 e^2+11 e f x+3 f^2 x^2\right )\right )\right )}{315 d^5 f^2}-\frac {2 (b c-a d)^2 (-d e+c f)^{5/2} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{11/2}} \] Input:
Integrate[((a + b*x)^2*(e + f*x)^(5/2))/(c + d*x),x]
Output:
(2*Sqrt[e + f*x]*(21*a^2*d^2*f^2*(15*c^2*f^2 - 5*c*d*f*(7*e + f*x) + d^2*( 23*e^2 + 11*e*f*x + 3*f^2*x^2)) + 6*a*b*d*f*(-105*c^3*f^3 + 15*d^3*(e + f* x)^3 + 35*c^2*d*f^2*(7*e + f*x) - 7*c*d^2*f*(23*e^2 + 11*e*f*x + 3*f^2*x^2 )) + b^2*(315*c^4*f^4 - 45*c*d^3*f*(e + f*x)^3 - 5*d^4*(2*e - 7*f*x)*(e + f*x)^3 - 105*c^3*d*f^3*(7*e + f*x) + 21*c^2*d^2*f^2*(23*e^2 + 11*e*f*x + 3 *f^2*x^2))))/(315*d^5*f^2) - (2*(b*c - a*d)^2*(-(d*e) + c*f)^(5/2)*ArcTan[ (Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(11/2)
Time = 0.39 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {99, 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.
| \(\displaystyle \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{d^2 f}+\frac {(e+f x)^{5/2} (a d-b c)^2}{d^2 (c+d x)}+\frac {b^2 (e+f x)^{7/2}}{d f}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (b c-a d)^2 (d e-c f)^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{11/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)^2}{d^5}+\frac {2 (e+f x)^{3/2} (b c-a d)^2 (d e-c f)}{3 d^4}+\frac {2 (e+f x)^{5/2} (b c-a d)^2}{5 d^3}-\frac {2 b (e+f x)^{7/2} (-2 a d f+b c f+b d e)}{7 d^2 f^2}+\frac {2 b^2 (e+f x)^{9/2}}{9 d f^2}\) |
Input:
Int[((a + b*x)^2*(e + f*x)^(5/2))/(c + d*x),x]
Output:
(2*(b*c - a*d)^2*(d*e - c*f)^2*Sqrt[e + f*x])/d^5 + (2*(b*c - a*d)^2*(d*e - c*f)*(e + f*x)^(3/2))/(3*d^4) + (2*(b*c - a*d)^2*(e + f*x)^(5/2))/(5*d^3 ) - (2*b*(b*d*e + b*c*f - 2*a*d*f)*(e + f*x)^(7/2))/(7*d^2*f^2) + (2*b^2*( e + f*x)^(9/2))/(9*d*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)^(5/2)*ArcTanh[(Sq rt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(11/2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Time = 0.63 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.41
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (-\left (\left (-\frac {2 \left (f x +e \right )^{3} \left (-\frac {7 f x}{2}+e \right ) d^{4}}{63}-\frac {c f \left (f x +e \right )^{3} d^{3}}{7}+\frac {23 c^{2} \left (\frac {3}{23} f^{2} x^{2}+\frac {11}{23} e f x +e^{2}\right ) f^{2} d^{2}}{15}-\frac {7 c^{3} f^{3} \left (\frac {f x}{7}+e \right ) d}{3}+c^{4} f^{4}\right ) b^{2}-2 a d f \left (-\frac {\left (f x +e \right )^{3} d^{3}}{7}+\frac {23 c \left (\frac {3}{23} f^{2} x^{2}+\frac {11}{23} e f x +e^{2}\right ) f \,d^{2}}{15}-\frac {7 c^{2} f^{2} \left (\frac {f x}{7}+e \right ) d}{3}+c^{3} f^{3}\right ) b +\left (\frac {\left (\frac {23}{3} e^{2}+f^{2} x^{2}+\frac {11}{3} e f x \right ) d^{2}}{5}-\frac {7 c f \left (\frac {f x}{7}+e \right ) d}{3}+c^{2} f^{2}\right ) a^{2} d^{2} f^{2}\right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}+f^{2} \left (c f -d e \right )^{3} \left (a d -b c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{2} d^{5}}\) | \(294\) |
| derivativedivides | \(\frac {\frac {2 \left (\frac {b^{2} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}+\frac {2 a b \,d^{4} f \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {b^{2} c \,d^{3} f \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {b^{2} d^{4} e \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{4} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 a b c \,d^{3} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c^{2} d^{2} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{2} c \,d^{3} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a^{2} d^{4} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 a b \,c^{2} d^{2} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 a b c \,d^{3} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c^{3} d \,f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c^{2} d^{2} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} c^{2} d^{2} f^{4} \sqrt {f x +e}-2 a^{2} c \,d^{3} e \,f^{3} \sqrt {f x +e}+a^{2} d^{4} e^{2} f^{2} \sqrt {f x +e}-2 a b \,c^{3} d \,f^{4} \sqrt {f x +e}+4 a b \,c^{2} d^{2} e \,f^{3} \sqrt {f x +e}-2 a b c \,d^{3} e^{2} f^{2} \sqrt {f x +e}+b^{2} c^{4} f^{4} \sqrt {f x +e}-2 b^{2} c^{3} d e \,f^{3} \sqrt {f x +e}+b^{2} c^{2} d^{2} e^{2} f^{2} \sqrt {f x +e}\right )}{d^{5}}-\frac {2 f^{2} \left (a^{2} c^{3} d^{2} f^{3}-3 a^{2} c^{2} d^{3} e \,f^{2}+3 a^{2} c \,d^{4} e^{2} f -a^{2} e^{3} d^{5}-2 a b \,c^{4} d \,f^{3}+6 a b \,c^{3} d^{2} e \,f^{2}-6 a b \,c^{2} d^{3} e^{2} f +2 a b c \,d^{4} e^{3}+c^{5} b^{2} f^{3}-3 b^{2} c^{4} d e \,f^{2}+3 b^{2} c^{3} d^{2} e^{2} f -b^{2} c^{2} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(627\) |
| default | \(\frac {\frac {2 \left (\frac {b^{2} \left (f x +e \right )^{\frac {9}{2}} d^{4}}{9}+\frac {2 a b \,d^{4} f \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {b^{2} c \,d^{3} f \left (f x +e \right )^{\frac {7}{2}}}{7}-\frac {b^{2} d^{4} e \left (f x +e \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{4} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {2 a b c \,d^{3} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {b^{2} c^{2} d^{2} f^{2} \left (f x +e \right )^{\frac {5}{2}}}{5}-\frac {a^{2} c \,d^{3} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {a^{2} d^{4} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {2 a b \,c^{2} d^{2} f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {2 a b c \,d^{3} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}-\frac {b^{2} c^{3} d \,f^{3} \left (f x +e \right )^{\frac {3}{2}}}{3}+\frac {b^{2} c^{2} d^{2} e \,f^{2} \left (f x +e \right )^{\frac {3}{2}}}{3}+a^{2} c^{2} d^{2} f^{4} \sqrt {f x +e}-2 a^{2} c \,d^{3} e \,f^{3} \sqrt {f x +e}+a^{2} d^{4} e^{2} f^{2} \sqrt {f x +e}-2 a b \,c^{3} d \,f^{4} \sqrt {f x +e}+4 a b \,c^{2} d^{2} e \,f^{3} \sqrt {f x +e}-2 a b c \,d^{3} e^{2} f^{2} \sqrt {f x +e}+b^{2} c^{4} f^{4} \sqrt {f x +e}-2 b^{2} c^{3} d e \,f^{3} \sqrt {f x +e}+b^{2} c^{2} d^{2} e^{2} f^{2} \sqrt {f x +e}\right )}{d^{5}}-\frac {2 f^{2} \left (a^{2} c^{3} d^{2} f^{3}-3 a^{2} c^{2} d^{3} e \,f^{2}+3 a^{2} c \,d^{4} e^{2} f -a^{2} e^{3} d^{5}-2 a b \,c^{4} d \,f^{3}+6 a b \,c^{3} d^{2} e \,f^{2}-6 a b \,c^{2} d^{3} e^{2} f +2 a b c \,d^{4} e^{3}+c^{5} b^{2} f^{3}-3 b^{2} c^{4} d e \,f^{2}+3 b^{2} c^{3} d^{2} e^{2} f -b^{2} c^{2} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(627\) |
| risch | \(\frac {2 \left (35 b^{2} f^{4} d^{4} x^{4}+90 a b \,d^{4} f^{4} x^{3}-45 b^{2} c \,d^{3} f^{4} x^{3}+95 b^{2} d^{4} e \,f^{3} x^{3}+63 a^{2} d^{4} f^{4} x^{2}-126 a b c \,d^{3} f^{4} x^{2}+270 a b \,d^{4} e \,f^{3} x^{2}+63 b^{2} c^{2} d^{2} f^{4} x^{2}-135 b^{2} c \,d^{3} e \,f^{3} x^{2}+75 b^{2} d^{4} e^{2} f^{2} x^{2}-105 a^{2} c \,d^{3} f^{4} x +231 a^{2} d^{4} e \,f^{3} x +210 a b \,c^{2} d^{2} f^{4} x -462 a b c \,d^{3} e \,f^{3} x +270 a b \,d^{4} e^{2} f^{2} x -105 b^{2} c^{3} d \,f^{4} x +231 b^{2} c^{2} d^{2} e \,f^{3} x -135 b^{2} c \,d^{3} e^{2} f^{2} x +5 b^{2} d^{4} e^{3} f x +315 a^{2} c^{2} d^{2} f^{4}-735 a^{2} c \,d^{3} e \,f^{3}+483 a^{2} d^{4} e^{2} f^{2}-630 a b \,c^{3} d \,f^{4}+1470 a b \,c^{2} d^{2} e \,f^{3}-966 a b c \,d^{3} e^{2} f^{2}+90 a b \,d^{4} e^{3} f +315 b^{2} c^{4} f^{4}-735 b^{2} c^{3} d e \,f^{3}+483 b^{2} c^{2} d^{2} e^{2} f^{2}-45 b^{2} c \,d^{3} e^{3} f -10 b^{2} d^{4} e^{4}\right ) \sqrt {f x +e}}{315 f^{2} d^{5}}-\frac {2 \left (a^{2} c^{3} d^{2} f^{3}-3 a^{2} c^{2} d^{3} e \,f^{2}+3 a^{2} c \,d^{4} e^{2} f -a^{2} e^{3} d^{5}-2 a b \,c^{4} d \,f^{3}+6 a b \,c^{3} d^{2} e \,f^{2}-6 a b \,c^{2} d^{3} e^{2} f +2 a b c \,d^{4} e^{3}+c^{5} b^{2} f^{3}-3 b^{2} c^{4} d e \,f^{2}+3 b^{2} c^{3} d^{2} e^{2} f -b^{2} c^{2} d^{3} e^{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{5} \sqrt {\left (c f -d e \right ) d}}\) | \(642\) |
Input:
int((b*x+a)^2*(f*x+e)^(5/2)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-2*(-((-2/63*(f*x+e)^3*(-7/2*f*x+e)*d^4-1/7*c*f*(f*x+e)^3*d^3+23/15*c^2*(3 /23*f^2*x^2+11/23*e*f*x+e^2)*f^2*d^2-7/3*c^3*f^3*(1/7*f*x+e)*d+c^4*f^4)*b^ 2-2*a*d*f*(-1/7*(f*x+e)^3*d^3+23/15*c*(3/23*f^2*x^2+11/23*e*f*x+e^2)*f*d^2 -7/3*c^2*f^2*(1/7*f*x+e)*d+c^3*f^3)*b+(1/5*(23/3*e^2+f^2*x^2+11/3*e*f*x)*d ^2-7/3*c*f*(1/7*f*x+e)*d+c^2*f^2)*a^2*d^2*f^2)*((c*f-d*e)*d)^(1/2)*(f*x+e) ^(1/2)+f^2*(c*f-d*e)^3*(a*d-b*c)^2*arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1 /2)))/((c*f-d*e)*d)^(1/2)/f^2/d^5
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (180) = 360\).
Time = 0.10 (sec) , antiderivative size = 1068, normalized size of antiderivative = 5.13 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a)^2*(f*x+e)^(5/2)/(d*x+c),x, algorithm="fricas")
Output:
[1/315*(315*((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e^2*f^2 - 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + (b^2*c^4 - 2*a*b*c^3*d + a^2*c^2*d^2) *f^4)*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f - 2*sqrt(f*x + e)*d*sqr t((d*e - c*f)/d))/(d*x + c)) + 2*(35*b^2*d^4*f^4*x^4 - 10*b^2*d^4*e^4 - 45 *(b^2*c*d^3 - 2*a*b*d^4)*e^3*f + 483*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4) *e^2*f^2 - 735*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + 315*(b^2*c^ 4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 + 5*(19*b^2*d^4*e*f^3 - 9*(b^2*c*d^3 - 2*a*b*d^4)*f^4)*x^3 + 3*(25*b^2*d^4*e^2*f^2 - 45*(b^2*c*d^3 - 2*a*b*d^4)*e *f^3 + 21*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^4)*x^2 + (5*b^2*d^4*e^3* f - 135*(b^2*c*d^3 - 2*a*b*d^4)*e^2*f^2 + 231*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e*f^3 - 105*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*f^4)*x)*sqrt (f*x + e))/(d^5*f^2), -2/315*(315*((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*e ^2*f^2 - 2*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + (b^2*c^4 - 2*a* b*c^3*d + a^2*c^2*d^2)*f^4)*sqrt(-(d*e - c*f)/d)*arctan(-sqrt(f*x + e)*d*s qrt(-(d*e - c*f)/d)/(d*e - c*f)) - (35*b^2*d^4*f^4*x^4 - 10*b^2*d^4*e^4 - 45*(b^2*c*d^3 - 2*a*b*d^4)*e^3*f + 483*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^ 4)*e^2*f^2 - 735*(b^2*c^3*d - 2*a*b*c^2*d^2 + a^2*c*d^3)*e*f^3 + 315*(b^2* c^4 - 2*a*b*c^3*d + a^2*c^2*d^2)*f^4 + 5*(19*b^2*d^4*e*f^3 - 9*(b^2*c*d^3 - 2*a*b*d^4)*f^4)*x^3 + 3*(25*b^2*d^4*e^2*f^2 - 45*(b^2*c*d^3 - 2*a*b*d^4) *e*f^3 + 21*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*f^4)*x^2 + (5*b^2*d^4...
Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (192) = 384\).
Time = 4.19 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} \left (e + f x\right )^{\frac {9}{2}}}{9 d f} + \frac {\left (e + f x\right )^{\frac {7}{2}} \cdot \left (2 a b d f - b^{2} c f - b^{2} d e\right )}{7 d^{2} f} + \frac {\left (e + f x\right )^{\frac {5}{2}} \left (a^{2} d^{2} f - 2 a b c d f + b^{2} c^{2} f\right )}{5 d^{3}} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (- a^{2} c d^{2} f^{2} + a^{2} d^{3} e f + 2 a b c^{2} d f^{2} - 2 a b c d^{2} e f - b^{2} c^{3} f^{2} + b^{2} c^{2} d e f\right )}{3 d^{4}} + \frac {\sqrt {e + f x} \left (a^{2} c^{2} d^{2} f^{3} - 2 a^{2} c d^{3} e f^{2} + a^{2} d^{4} e^{2} f - 2 a b c^{3} d f^{3} + 4 a b c^{2} d^{2} e f^{2} - 2 a b c d^{3} e^{2} f + b^{2} c^{4} f^{3} - 2 b^{2} c^{3} d e f^{2} + b^{2} c^{2} d^{2} e^{2} f\right )}{d^{5}} - \frac {f \left (a d - b c\right )^{2} \left (c f - d e\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{6} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {5}{2}} \left (\frac {b^{2} x^{2}}{2 d} + \frac {x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac {\left (a d - b c\right )^{2} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)**2*(f*x+e)**(5/2)/(d*x+c),x)
Output:
Piecewise((2*(b**2*(e + f*x)**(9/2)/(9*d*f) + (e + f*x)**(7/2)*(2*a*b*d*f - b**2*c*f - b**2*d*e)/(7*d**2*f) + (e + f*x)**(5/2)*(a**2*d**2*f - 2*a*b* c*d*f + b**2*c**2*f)/(5*d**3) + (e + f*x)**(3/2)*(-a**2*c*d**2*f**2 + a**2 *d**3*e*f + 2*a*b*c**2*d*f**2 - 2*a*b*c*d**2*e*f - b**2*c**3*f**2 + b**2*c **2*d*e*f)/(3*d**4) + sqrt(e + f*x)*(a**2*c**2*d**2*f**3 - 2*a**2*c*d**3*e *f**2 + a**2*d**4*e**2*f - 2*a*b*c**3*d*f**3 + 4*a*b*c**2*d**2*e*f**2 - 2* a*b*c*d**3*e**2*f + b**2*c**4*f**3 - 2*b**2*c**3*d*e*f**2 + b**2*c**2*d**2 *e**2*f)/d**5 - f*(a*d - b*c)**2*(c*f - d*e)**3*atan(sqrt(e + f*x)/sqrt((c *f - d*e)/d))/(d**6*sqrt((c*f - d*e)/d)))/f, Ne(f, 0)), (e**(5/2)*(b**2*x* *2/(2*d) + x*(2*a*b*d - b**2*c)/d**2 + (a*d - b*c)**2*Piecewise((x/c, Eq(d , 0)), (log(c + d*x)/d, True))/d**2), True))
Exception generated. \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a)^2*(f*x+e)^(5/2)/(d*x+c),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (180) = 360\).
Time = 0.13 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.11 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx=\frac {2 \, {\left (b^{2} c^{2} d^{3} e^{3} - 2 \, a b c d^{4} e^{3} + a^{2} d^{5} e^{3} - 3 \, b^{2} c^{3} d^{2} e^{2} f + 6 \, a b c^{2} d^{3} e^{2} f - 3 \, a^{2} c d^{4} e^{2} f + 3 \, b^{2} c^{4} d e f^{2} - 6 \, a b c^{3} d^{2} e f^{2} + 3 \, a^{2} c^{2} d^{3} e f^{2} - b^{2} c^{5} f^{3} + 2 \, a b c^{4} d f^{3} - a^{2} c^{3} d^{2} f^{3}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{5}} + \frac {2 \, {\left (35 \, {\left (f x + e\right )}^{\frac {9}{2}} b^{2} d^{8} f^{16} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} d^{8} e f^{16} - 45 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} c d^{7} f^{17} + 90 \, {\left (f x + e\right )}^{\frac {7}{2}} a b d^{8} f^{17} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} c^{2} d^{6} f^{18} - 126 \, {\left (f x + e\right )}^{\frac {5}{2}} a b c d^{7} f^{18} + 63 \, {\left (f x + e\right )}^{\frac {5}{2}} a^{2} d^{8} f^{18} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} d^{6} e f^{18} - 210 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c d^{7} e f^{18} + 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{8} e f^{18} + 315 \, \sqrt {f x + e} b^{2} c^{2} d^{6} e^{2} f^{18} - 630 \, \sqrt {f x + e} a b c d^{7} e^{2} f^{18} + 315 \, \sqrt {f x + e} a^{2} d^{8} e^{2} f^{18} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{3} d^{5} f^{19} + 210 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c^{2} d^{6} f^{19} - 105 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} c d^{7} f^{19} - 630 \, \sqrt {f x + e} b^{2} c^{3} d^{5} e f^{19} + 1260 \, \sqrt {f x + e} a b c^{2} d^{6} e f^{19} - 630 \, \sqrt {f x + e} a^{2} c d^{7} e f^{19} + 315 \, \sqrt {f x + e} b^{2} c^{4} d^{4} f^{20} - 630 \, \sqrt {f x + e} a b c^{3} d^{5} f^{20} + 315 \, \sqrt {f x + e} a^{2} c^{2} d^{6} f^{20}\right )}}{315 \, d^{9} f^{18}} \] Input:
integrate((b*x+a)^2*(f*x+e)^(5/2)/(d*x+c),x, algorithm="giac")
Output:
2*(b^2*c^2*d^3*e^3 - 2*a*b*c*d^4*e^3 + a^2*d^5*e^3 - 3*b^2*c^3*d^2*e^2*f + 6*a*b*c^2*d^3*e^2*f - 3*a^2*c*d^4*e^2*f + 3*b^2*c^4*d*e*f^2 - 6*a*b*c^3*d ^2*e*f^2 + 3*a^2*c^2*d^3*e*f^2 - b^2*c^5*f^3 + 2*a*b*c^4*d*f^3 - a^2*c^3*d ^2*f^3)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f) *d^5) + 2/315*(35*(f*x + e)^(9/2)*b^2*d^8*f^16 - 45*(f*x + e)^(7/2)*b^2*d^ 8*e*f^16 - 45*(f*x + e)^(7/2)*b^2*c*d^7*f^17 + 90*(f*x + e)^(7/2)*a*b*d^8* f^17 + 63*(f*x + e)^(5/2)*b^2*c^2*d^6*f^18 - 126*(f*x + e)^(5/2)*a*b*c*d^7 *f^18 + 63*(f*x + e)^(5/2)*a^2*d^8*f^18 + 105*(f*x + e)^(3/2)*b^2*c^2*d^6* e*f^18 - 210*(f*x + e)^(3/2)*a*b*c*d^7*e*f^18 + 105*(f*x + e)^(3/2)*a^2*d^ 8*e*f^18 + 315*sqrt(f*x + e)*b^2*c^2*d^6*e^2*f^18 - 630*sqrt(f*x + e)*a*b* c*d^7*e^2*f^18 + 315*sqrt(f*x + e)*a^2*d^8*e^2*f^18 - 105*(f*x + e)^(3/2)* b^2*c^3*d^5*f^19 + 210*(f*x + e)^(3/2)*a*b*c^2*d^6*f^19 - 105*(f*x + e)^(3 /2)*a^2*c*d^7*f^19 - 630*sqrt(f*x + e)*b^2*c^3*d^5*e*f^19 + 1260*sqrt(f*x + e)*a*b*c^2*d^6*e*f^19 - 630*sqrt(f*x + e)*a^2*c*d^7*e*f^19 + 315*sqrt(f* x + e)*b^2*c^4*d^4*f^20 - 630*sqrt(f*x + e)*a*b*c^3*d^5*f^20 + 315*sqrt(f* x + e)*a^2*c^2*d^6*f^20)/(d^9*f^18)
Time = 1.15 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.89 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx={\left (e+f\,x\right )}^{5/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{5\,d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d\,f^2}\right )-{\left (e+f\,x\right )}^{7/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{7\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{7\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{9/2}}{9\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{5/2}}{-a^2\,c^3\,d^2\,f^3+3\,a^2\,c^2\,d^3\,e\,f^2-3\,a^2\,c\,d^4\,e^2\,f+a^2\,d^5\,e^3+2\,a\,b\,c^4\,d\,f^3-6\,a\,b\,c^3\,d^2\,e\,f^2+6\,a\,b\,c^2\,d^3\,e^2\,f-2\,a\,b\,c\,d^4\,e^3-b^2\,c^5\,f^3+3\,b^2\,c^4\,d\,e\,f^2-3\,b^2\,c^3\,d^2\,e^2\,f+b^2\,c^2\,d^3\,e^3}\right )\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{5/2}}{d^{11/2}}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d\,f^2}+\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,{\left (c\,f^3-d\,e\,f^2\right )}^2}{d^2\,f^4} \] Input:
int(((e + f*x)^(5/2)*(a + b*x)^2)/(c + d*x),x)
Output:
(e + f*x)^(5/2)*((2*(a*f - b*e)^2)/(5*d*f^2) + (((4*b^2*e - 4*a*b*f)/(d*f^ 2) + (2*b^2*(c*f^3 - d*e*f^2))/(d^2*f^4))*(c*f^3 - d*e*f^2))/(5*d*f^2)) - (e + f*x)^(7/2)*((4*b^2*e - 4*a*b*f)/(7*d*f^2) + (2*b^2*(c*f^3 - d*e*f^2)) /(7*d^2*f^4)) + (2*b^2*(e + f*x)^(9/2))/(9*d*f^2) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^2*(c*f - d*e)^(5/2))/(a^2*d^5*e^3 - b^2*c^5*f^3 - a ^2*c^3*d^2*f^3 + b^2*c^2*d^3*e^3 - 2*a*b*c*d^4*e^3 + 2*a*b*c^4*d*f^3 - 3*a ^2*c*d^4*e^2*f + 3*b^2*c^4*d*e*f^2 + 3*a^2*c^2*d^3*e*f^2 - 3*b^2*c^3*d^2*e ^2*f + 6*a*b*c^2*d^3*e^2*f - 6*a*b*c^3*d^2*e*f^2))*(a*d - b*c)^2*(c*f - d* e)^(5/2))/d^(11/2) - ((e + f*x)^(3/2)*((2*(a*f - b*e)^2)/(d*f^2) + (((4*b^ 2*e - 4*a*b*f)/(d*f^2) + (2*b^2*(c*f^3 - d*e*f^2))/(d^2*f^4))*(c*f^3 - d*e *f^2))/(d*f^2))*(c*f^3 - d*e*f^2))/(3*d*f^2) + ((e + f*x)^(1/2)*((2*(a*f - b*e)^2)/(d*f^2) + (((4*b^2*e - 4*a*b*f)/(d*f^2) + (2*b^2*(c*f^3 - d*e*f^2 ))/(d^2*f^4))*(c*f^3 - d*e*f^2))/(d*f^2))*(c*f^3 - d*e*f^2)^2)/(d^2*f^4)
Time = 0.16 (sec) , antiderivative size = 1068, normalized size of antiderivative = 5.13 \[ \int \frac {(a+b x)^2 (e+f x)^{5/2}}{c+d x} \, dx =\text {Too large to display} \] Input:
int((b*x+a)^2*(f*x+e)^(5/2)/(d*x+c),x)
Output:
(2*( - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c* f - d*e)))*a**2*c**2*d**2*f**4 + 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d**3*e*f**3 - 315*sqrt(d)*sqrt (c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*d**4*e* *2*f**2 + 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt (c*f - d*e)))*a*b*c**3*d*f**4 - 1260*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c**2*d**2*e*f**3 + 630*sqrt(d)*sq rt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c*d**3 *e**2*f**2 - 315*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s qrt(c*f - d*e)))*b**2*c**4*f**4 + 630*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**2*c**3*d*e*f**3 - 315*sqrt(d)*sqr t(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**2*c**2*d **2*e**2*f**2 + 315*sqrt(e + f*x)*a**2*c**2*d**3*f**4 - 735*sqrt(e + f*x)* a**2*c*d**4*e*f**3 - 105*sqrt(e + f*x)*a**2*c*d**4*f**4*x + 483*sqrt(e + f *x)*a**2*d**5*e**2*f**2 + 231*sqrt(e + f*x)*a**2*d**5*e*f**3*x + 63*sqrt(e + f*x)*a**2*d**5*f**4*x**2 - 630*sqrt(e + f*x)*a*b*c**3*d**2*f**4 + 1470* sqrt(e + f*x)*a*b*c**2*d**3*e*f**3 + 210*sqrt(e + f*x)*a*b*c**2*d**3*f**4* x - 966*sqrt(e + f*x)*a*b*c*d**4*e**2*f**2 - 462*sqrt(e + f*x)*a*b*c*d**4* e*f**3*x - 126*sqrt(e + f*x)*a*b*c*d**4*f**4*x**2 + 90*sqrt(e + f*x)*a*b*d **5*e**3*f + 270*sqrt(e + f*x)*a*b*d**5*e**2*f**2*x + 270*sqrt(e + f*x)...