Integrand size = 24, antiderivative size = 172 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\frac {2 (b c-a d)^2 (d e-c f) \sqrt {e+f x}}{d^4}+\frac {2 (b c-a d)^2 (e+f x)^{3/2}}{3 d^3}-\frac {2 b (b d e+b c f-2 a d f) (e+f x)^{5/2}}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}-\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}} \]
2/3*(-a*d+b*c)^2*(f*x+e)^(3/2)/d^3-2/5*b*(-2*a*d*f+b*c*f+b*d*e)*(f*x+e)^(5 /2)/d^2/f^2+2/7*b^2*(f*x+e)^(7/2)/d/f^2-2*(-a*d+b*c)^2*(-c*f+d*e)^(3/2)*ar ctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(9/2)+2*(-a*d+b*c)^2*(-c*f +d*e)*(f*x+e)^(1/2)/d^4
Time = 0.30 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\frac {2 \sqrt {e+f x} \left (35 a^2 d^2 f^2 (4 d e-3 c f+d f x)+14 a b d f \left (15 c^2 f^2+3 d^2 (e+f x)^2-5 c d f (4 e+f x)\right )+b^2 \left (-105 c^3 f^3-21 c d^2 f (e+f x)^2-3 d^3 (2 e-5 f x) (e+f x)^2+35 c^2 d f^2 (4 e+f x)\right )\right )}{105 d^4 f^2}+\frac {2 (b c-a d)^2 (-d e+c f)^{3/2} \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{9/2}} \]
(2*Sqrt[e + f*x]*(35*a^2*d^2*f^2*(4*d*e - 3*c*f + d*f*x) + 14*a*b*d*f*(15* c^2*f^2 + 3*d^2*(e + f*x)^2 - 5*c*d*f*(4*e + f*x)) + b^2*(-105*c^3*f^3 - 2 1*c*d^2*f*(e + f*x)^2 - 3*d^3*(2*e - 5*f*x)*(e + f*x)^2 + 35*c^2*d*f^2*(4* e + f*x))))/(105*d^4*f^2) + (2*(b*c - a*d)^2*(-(d*e) + c*f)^(3/2)*ArcTan[( Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/d^(9/2)
Time = 0.32 (sec) , antiderivative size = 172, 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.
| \(\displaystyle \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \int \left (-\frac {b (e+f x)^{3/2} (-2 a d f+b c f+b d e)}{d^2 f}+\frac {(e+f x)^{3/2} (a d-b c)^2}{d^2 (c+d x)}+\frac {b^2 (e+f x)^{5/2}}{d f}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 (b c-a d)^2 (d e-c f)^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{9/2}}+\frac {2 \sqrt {e+f x} (b c-a d)^2 (d e-c f)}{d^4}+\frac {2 (e+f x)^{3/2} (b c-a d)^2}{3 d^3}-\frac {2 b (e+f x)^{5/2} (-2 a d f+b c f+b d e)}{5 d^2 f^2}+\frac {2 b^2 (e+f x)^{7/2}}{7 d f^2}\) |
(2*(b*c - a*d)^2*(d*e - c*f)*Sqrt[e + f*x])/d^4 + (2*(b*c - a*d)^2*(e + f* x)^(3/2))/(3*d^3) - (2*b*(b*d*e + b*c*f - 2*a*d*f)*(e + f*x)^(5/2))/(5*d^2 *f^2) + (2*b^2*(e + f*x)^(7/2))/(7*d*f^2) - (2*(b*c - a*d)^2*(d*e - c*f)^( 3/2)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(9/2)
3.18.75.3.1 Defintions of rubi rules used
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 = 1.07 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\sqrt {\left (c f -d e \right ) d}\, \left (\left (\frac {2 \left (-\frac {5 f x}{2}+e \right ) \left (f x +e \right )^{2} d^{3}}{35}+\frac {c f \left (f x +e \right )^{2} d^{2}}{5}-\frac {4 \left (\frac {f x}{4}+e \right ) f^{2} c^{2} d}{3}+c^{3} f^{3}\right ) b^{2}-2 f d \left (\frac {\left (f x +e \right )^{2} d^{2}}{5}-\frac {4 \left (\frac {f x}{4}+e \right ) f c d}{3}+c^{2} f^{2}\right ) a b +f^{2} d^{2} a^{2} \left (\frac {\left (-f x -4 e \right ) d}{3}+c f \right )\right ) \sqrt {f x +e}-f^{2} \left (c f -d e \right )^{2} \left (a d -b c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )\right )}{\sqrt {\left (c f -d e \right ) d}\, f^{2} d^{4}}\) | \(211\) |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {b^{2} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {\left (-\left (a d f -b c f \right ) b \,d^{2}+b d \left (-a \,d^{2} f +d^{2} e b \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a \,d^{2} f +d^{2} e b \right )+b d \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right ) \sqrt {f x +e}\right )}{d^{4}}+\frac {2 f^{2} \left (a^{2} c^{2} d^{2} f^{2}-2 a^{2} c \,d^{3} e f +a^{2} e^{2} d^{4}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+c^{4} b^{2} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(333\) |
| default | \(\frac {-\frac {2 \left (-\frac {b^{2} \left (f x +e \right )^{\frac {7}{2}} d^{3}}{7}+\frac {\left (-\left (a d f -b c f \right ) b \,d^{2}+b d \left (-a \,d^{2} f +d^{2} e b \right )\right ) \left (f x +e \right )^{\frac {5}{2}}}{5}+\frac {\left (\left (a d f -b c f \right ) \left (-a \,d^{2} f +d^{2} e b \right )+b d \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right )\right ) \left (f x +e \right )^{\frac {3}{2}}}{3}+\left (a d f -b c f \right ) \left (a c d \,f^{2}-a \,d^{2} e f -b \,c^{2} f^{2}+b c d e f \right ) \sqrt {f x +e}\right )}{d^{4}}+\frac {2 f^{2} \left (a^{2} c^{2} d^{2} f^{2}-2 a^{2} c \,d^{3} e f +a^{2} e^{2} d^{4}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+c^{4} b^{2} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}}{f^{2}}\) | \(333\) |
| risch | \(-\frac {2 \left (-15 b^{2} f^{3} d^{3} x^{3}-42 a b \,d^{3} f^{3} x^{2}+21 b^{2} c \,d^{2} f^{3} x^{2}-24 b^{2} d^{3} e \,f^{2} x^{2}-35 a^{2} d^{3} f^{3} x +70 a b c \,d^{2} f^{3} x -84 a b \,d^{3} e \,f^{2} x -35 b^{2} c^{2} d \,f^{3} x +42 b^{2} c \,d^{2} e \,f^{2} x -3 b^{2} d^{3} e^{2} f x +105 a^{2} c \,d^{2} f^{3}-140 a^{2} d^{3} e \,f^{2}-210 a b \,c^{2} d \,f^{3}+280 a b c \,d^{2} e \,f^{2}-42 a b \,d^{3} e^{2} f +105 b^{2} c^{3} f^{3}-140 b^{2} c^{2} d e \,f^{2}+21 b^{2} c \,d^{2} e^{2} f +6 b^{2} d^{3} e^{3}\right ) \sqrt {f x +e}}{105 f^{2} d^{4}}+\frac {2 \left (a^{2} c^{2} d^{2} f^{2}-2 a^{2} c \,d^{3} e f +a^{2} e^{2} d^{4}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+c^{4} b^{2} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d^{4} \sqrt {\left (c f -d e \right ) d}}\) | \(399\) |
-2/((c*f-d*e)*d)^(1/2)*(((c*f-d*e)*d)^(1/2)*((2/35*(-5/2*f*x+e)*(f*x+e)^2* d^3+1/5*c*f*(f*x+e)^2*d^2-4/3*(1/4*f*x+e)*f^2*c^2*d+c^3*f^3)*b^2-2*f*d*(1/ 5*(f*x+e)^2*d^2-4/3*(1/4*f*x+e)*f*c*d+c^2*f^2)*a*b+f^2*d^2*a^2*(1/3*(-f*x- 4*e)*d+c*f))*(f*x+e)^(1/2)-f^2*(c*f-d*e)^2*(a*d-b*c)^2*arctan(d*(f*x+e)^(1 /2)/((c*f-d*e)*d)^(1/2)))/f^2/d^4
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (148) = 296\).
Time = 0.24 (sec) , antiderivative size = 694, normalized size of antiderivative = 4.03 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\left [-\frac {105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt {\frac {d e - c f}{d}} \log \left (\frac {d f x + 2 \, d e - c f + 2 \, \sqrt {f x + e} d \sqrt {\frac {d e - c f}{d}}}{d x + c}\right ) - 2 \, {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (8 \, b^{2} d^{3} e f^{2} - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} + {\left (3 \, b^{2} d^{3} e^{2} f - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}}{105 \, d^{4} f^{2}}, -\frac {2 \, {\left (105 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3}\right )} \sqrt {-\frac {d e - c f}{d}} \arctan \left (-\frac {\sqrt {f x + e} d \sqrt {-\frac {d e - c f}{d}}}{d e - c f}\right ) - {\left (15 \, b^{2} d^{3} f^{3} x^{3} - 6 \, b^{2} d^{3} e^{3} - 21 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e^{2} f + 140 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} e f^{2} - 105 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} f^{3} + 3 \, {\left (8 \, b^{2} d^{3} e f^{2} - 7 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} f^{3}\right )} x^{2} + {\left (3 \, b^{2} d^{3} e^{2} f - 42 \, {\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} e f^{2} + 35 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} f^{3}\right )} x\right )} \sqrt {f x + e}\right )}}{105 \, d^{4} f^{2}}\right ] \]
[-1/105*(105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - (b^2*c^3 - 2*a*b *c^2*d + a^2*c*d^2)*f^3)*sqrt((d*e - c*f)/d)*log((d*f*x + 2*d*e - c*f + 2* sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 2*(15*b^2*d^3*f^3*x^3 - 6*b^2*d^3*e^3 - 21*(b^2*c*d^2 - 2*a*b*d^3)*e^2*f + 140*(b^2*c^2*d - 2*a*b* c*d^2 + a^2*d^3)*e*f^2 - 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3 + 3*( 8*b^2*d^3*e*f^2 - 7*(b^2*c*d^2 - 2*a*b*d^3)*f^3)*x^2 + (3*b^2*d^3*e^2*f - 42*(b^2*c*d^2 - 2*a*b*d^3)*e*f^2 + 35*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)* f^3)*x)*sqrt(f*x + e))/(d^4*f^2), -2/105*(105*((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - (b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2)*f^3)*sqrt(-(d*e - c*f )/d)*arctan(-sqrt(f*x + e)*d*sqrt(-(d*e - c*f)/d)/(d*e - c*f)) - (15*b^2*d ^3*f^3*x^3 - 6*b^2*d^3*e^3 - 21*(b^2*c*d^2 - 2*a*b*d^3)*e^2*f + 140*(b^2*c ^2*d - 2*a*b*c*d^2 + a^2*d^3)*e*f^2 - 105*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d ^2)*f^3 + 3*(8*b^2*d^3*e*f^2 - 7*(b^2*c*d^2 - 2*a*b*d^3)*f^3)*x^2 + (3*b^2 *d^3*e^2*f - 42*(b^2*c*d^2 - 2*a*b*d^3)*e*f^2 + 35*(b^2*c^2*d - 2*a*b*c*d^ 2 + a^2*d^3)*f^3)*x)*sqrt(f*x + e))/(d^4*f^2)]
Time = 3.25 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} \left (e + f x\right )^{\frac {7}{2}}}{7 d f} + \frac {\left (e + f x\right )^{\frac {5}{2}} \cdot \left (2 a b d f - b^{2} c f - b^{2} d e\right )}{5 d^{2} f} + \frac {\left (e + f x\right )^{\frac {3}{2}} \left (a^{2} d^{2} f - 2 a b c d f + b^{2} c^{2} f\right )}{3 d^{3}} + \frac {\sqrt {e + f x} \left (- a^{2} c d^{2} f^{2} + a^{2} d^{3} e f + 2 a b c^{2} d f^{2} - 2 a b c d^{2} e f - b^{2} c^{3} f^{2} + b^{2} c^{2} d e f\right )}{d^{4}} + \frac {f \left (a d - b c\right )^{2} \left (c f - d e\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {e + f x}}{\sqrt {\frac {c f - d e}{d}}} \right )}}{d^{5} \sqrt {\frac {c f - d e}{d}}}\right )}{f} & \text {for}\: f \neq 0 \\e^{\frac {3}{2}} \left (\frac {b^{2} x^{2}}{2 d} + \frac {x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac {\left (a d - b c\right )^{2} \left (\begin {cases} \frac {x}{c} & \text {for}\: d = 0 \\\frac {\log {\left (c + d x \right )}}{d} & \text {otherwise} \end {cases}\right )}{d^{2}}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*(b**2*(e + f*x)**(7/2)/(7*d*f) + (e + f*x)**(5/2)*(2*a*b*d*f - b**2*c*f - b**2*d*e)/(5*d**2*f) + (e + f*x)**(3/2)*(a**2*d**2*f - 2*a*b* c*d*f + b**2*c**2*f)/(3*d**3) + sqrt(e + f*x)*(-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)/d**4 + f*(a*d - b*c)**2*(c*f - d*e)**2*atan(sqrt(e + f*x)/sqrt((c* f - d*e)/d))/(d**5*sqrt((c*f - d*e)/d)))/f, Ne(f, 0)), (e**(3/2)*(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. \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\text {Exception raised: ValueError} \]
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. 404 vs. \(2 (148) = 296\).
Time = 0.28 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.35 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx=\frac {2 \, {\left (b^{2} c^{2} d^{2} e^{2} - 2 \, a b c d^{3} e^{2} + a^{2} d^{4} e^{2} - 2 \, b^{2} c^{3} d e f + 4 \, a b c^{2} d^{2} e f - 2 \, a^{2} c d^{3} e f + b^{2} c^{4} f^{2} - 2 \, a b c^{3} d f^{2} + a^{2} c^{2} d^{2} f^{2}\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{\sqrt {-d^{2} e + c d f} d^{4}} + \frac {2 \, {\left (15 \, {\left (f x + e\right )}^{\frac {7}{2}} b^{2} d^{6} f^{12} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} d^{6} e f^{12} - 21 \, {\left (f x + e\right )}^{\frac {5}{2}} b^{2} c d^{5} f^{13} + 42 \, {\left (f x + e\right )}^{\frac {5}{2}} a b d^{6} f^{13} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} d^{4} f^{14} - 70 \, {\left (f x + e\right )}^{\frac {3}{2}} a b c d^{5} f^{14} + 35 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{6} f^{14} + 105 \, \sqrt {f x + e} b^{2} c^{2} d^{4} e f^{14} - 210 \, \sqrt {f x + e} a b c d^{5} e f^{14} + 105 \, \sqrt {f x + e} a^{2} d^{6} e f^{14} - 105 \, \sqrt {f x + e} b^{2} c^{3} d^{3} f^{15} + 210 \, \sqrt {f x + e} a b c^{2} d^{4} f^{15} - 105 \, \sqrt {f x + e} a^{2} c d^{5} f^{15}\right )}}{105 \, d^{7} f^{14}} \]
2*(b^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 + a^2*d^4*e^2 - 2*b^2*c^3*d*e*f + 4*a *b*c^2*d^2*e*f - 2*a^2*c*d^3*e*f + b^2*c^4*f^2 - 2*a*b*c^3*d*f^2 + a^2*c^2 *d^2*f^2)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d* f)*d^4) + 2/105*(15*(f*x + e)^(7/2)*b^2*d^6*f^12 - 21*(f*x + e)^(5/2)*b^2* d^6*e*f^12 - 21*(f*x + e)^(5/2)*b^2*c*d^5*f^13 + 42*(f*x + e)^(5/2)*a*b*d^ 6*f^13 + 35*(f*x + e)^(3/2)*b^2*c^2*d^4*f^14 - 70*(f*x + e)^(3/2)*a*b*c*d^ 5*f^14 + 35*(f*x + e)^(3/2)*a^2*d^6*f^14 + 105*sqrt(f*x + e)*b^2*c^2*d^4*e *f^14 - 210*sqrt(f*x + e)*a*b*c*d^5*e*f^14 + 105*sqrt(f*x + e)*a^2*d^6*e*f ^14 - 105*sqrt(f*x + e)*b^2*c^3*d^3*f^15 + 210*sqrt(f*x + e)*a*b*c^2*d^4*f ^15 - 105*sqrt(f*x + e)*a^2*c*d^5*f^15)/(d^7*f^14)
Time = 1.39 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.55 \[ \int \frac {(a+b x)^2 (e+f x)^{3/2}}{c+d x} \, dx={\left (e+f\,x\right )}^{3/2}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{3\,d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{3\,d\,f^2}\right )-{\left (e+f\,x\right )}^{5/2}\,\left (\frac {4\,b^2\,e-4\,a\,b\,f}{5\,d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{5\,d^2\,f^4}\right )+\frac {2\,b^2\,{\left (e+f\,x\right )}^{7/2}}{7\,d\,f^2}+\frac {2\,\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{a^2\,c^2\,d^2\,f^2-2\,a^2\,c\,d^3\,e\,f+a^2\,d^4\,e^2-2\,a\,b\,c^3\,d\,f^2+4\,a\,b\,c^2\,d^2\,e\,f-2\,a\,b\,c\,d^3\,e^2+b^2\,c^4\,f^2-2\,b^2\,c^3\,d\,e\,f+b^2\,c^2\,d^2\,e^2}\right )\,{\left (a\,d-b\,c\right )}^2\,{\left (c\,f-d\,e\right )}^{3/2}}{d^{9/2}}-\frac {\sqrt {e+f\,x}\,\left (\frac {2\,{\left (a\,f-b\,e\right )}^2}{d\,f^2}+\frac {\left (\frac {4\,b^2\,e-4\,a\,b\,f}{d\,f^2}+\frac {2\,b^2\,\left (c\,f^3-d\,e\,f^2\right )}{d^2\,f^4}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2}\right )\,\left (c\,f^3-d\,e\,f^2\right )}{d\,f^2} \]
(e + f*x)^(3/2)*((2*(a*f - b*e)^2)/(3*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))/(3*d*f^2)) - (e + f*x)^(5/2)*((4*b^2*e - 4*a*b*f)/(5*d*f^2) + (2*b^2*(c*f^3 - d*e*f^2)) /(5*d^2*f^4)) + (2*b^2*(e + f*x)^(7/2))/(7*d*f^2) + (2*atan((d^(1/2)*(e + f*x)^(1/2)*(a*d - b*c)^2*(c*f - d*e)^(3/2))/(a^2*d^4*e^2 + b^2*c^4*f^2 + a ^2*c^2*d^2*f^2 + b^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 - 2*a*b*c^3*d*f^2 - 2*a ^2*c*d^3*e*f - 2*b^2*c^3*d*e*f + 4*a*b*c^2*d^2*e*f))*(a*d - b*c)^2*(c*f - d*e)^(3/2))/d^(9/2) - ((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))/(d*f^2)