\(\int (d+e x)^{5/2} (a+b x+c x^2)^2 \, dx\) [2275]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

[Out]

2/7*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(7/2)/e^5-4/9*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(9/2)/e^5+2/11*(6*c^2
*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^(11/2)/e^5-4/13*c*(-b*e+2*c*d)*(e*x+d)^(13/2)/e^5+2/15*c^2*(e*x+d)^(1
5/2)/e^5

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{11/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{11 e^5}-\frac {4 (d+e x)^{9/2} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{9 e^5}+\frac {2 (d+e x)^{7/2} \left (a e^2-b d e+c d^2\right )^2}{7 e^5}-\frac {4 c (d+e x)^{13/2} (2 c d-b e)}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \]

[In]

Int[(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]

[Out]

(2*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(9/
2))/(9*e^5) + (2*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(11/2))/(11*e^5) - (4*c*(2*c*d - b*e)*(
d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5)

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{5/2}}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{9/2}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{11/2}}{e^4}+\frac {c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx \\ & = \frac {2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{11/2}}{11 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{13/2}}{13 e^5}+\frac {2 c^2 (d+e x)^{15/2}}{15 e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.04 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (c^2 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+65 e^2 \left (99 a^2 e^2+22 a b e (-2 d+7 e x)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )-10 c e \left (-13 a e \left (8 d^2-28 d e x+63 e^2 x^2\right )+3 b \left (16 d^3-56 d^2 e x+126 d e^2 x^2-231 e^3 x^3\right )\right )\right )}{45045 e^5} \]

[In]

Integrate[(d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x]

[Out]

(2*(d + e*x)^(7/2)*(c^2*(128*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + 65*e^2*(9
9*a^2*e^2 + 22*a*b*e*(-2*d + 7*e*x) + b^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)) - 10*c*e*(-13*a*e*(8*d^2 - 28*d*e*x
+ 63*e^2*x^2) + 3*b*(16*d^3 - 56*d^2*e*x + 126*d*e^2*x^2 - 231*e^3*x^3))))/(45045*e^5)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(136\)
default	\(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {4 c \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) c +\left (b e -2 c d \right )^{2}\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (b e -2 c d \right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a \,e^{2}-b d e +c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{5}}\)	\(136\)
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (\left (\frac {7 c^{2} x^{4}}{15}+\frac {14 \left (\frac {11 b x}{13}+a \right ) x^{2} c}{11}+\frac {7 b^{2} x^{2}}{11}+\frac {14 a b x}{9}+a^{2}\right ) e^{4}-\frac {4 \left (\frac {42 c^{2} x^{3}}{65}+\frac {14 x \left (\frac {27 b x}{26}+a \right ) c}{11}+b \left (\frac {7 b x}{11}+a \right )\right ) d \,e^{3}}{9}+\frac {16 \left (\frac {63 c^{2} x^{2}}{65}+\left (\frac {21 b x}{13}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}}{99}-\frac {32 c \left (\frac {14 c x}{15}+b \right ) d^{3} e}{429}+\frac {128 c^{2} d^{4}}{6435}\right )}{7 e^{5}}\)	\(139\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 c^{2} x^{4} e^{4}+6930 b c \,e^{4} x^{3}-1848 c^{2} d \,e^{3} x^{3}+8190 a c \,e^{4} x^{2}+4095 b^{2} e^{4} x^{2}-3780 b c d \,e^{3} x^{2}+1008 c^{2} d^{2} e^{2} x^{2}+10010 a b \,e^{4} x -3640 a c d \,e^{3} x -1820 b^{2} d \,e^{3} x +1680 b c \,d^{2} e^{2} x -448 c^{2} d^{3} e x +6435 a^{2} e^{4}-2860 a b d \,e^{3}+1040 a c \,d^{2} e^{2}+520 b^{2} d^{2} e^{2}-480 b c \,d^{3} e +128 c^{2} d^{4}\right )}{45045 e^{5}}\)	\(194\)
trager	\(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(433\)
risch	\(\frac {2 \left (3003 c^{2} e^{7} x^{7}+6930 b c \,e^{7} x^{6}+7161 c^{2} d \,e^{6} x^{6}+8190 a c \,e^{7} x^{5}+4095 b^{2} e^{7} x^{5}+17010 b c d \,e^{6} x^{5}+4473 c^{2} d^{2} e^{5} x^{5}+10010 a b \,e^{7} x^{4}+20930 a c d \,e^{6} x^{4}+10465 b^{2} d \,e^{6} x^{4}+11130 b c \,d^{2} e^{5} x^{4}+35 c^{2} d^{3} e^{4} x^{4}+6435 a^{2} e^{7} x^{3}+27170 a b d \,e^{6} x^{3}+14690 a c \,d^{2} e^{5} x^{3}+7345 b^{2} d^{2} e^{5} x^{3}+150 b c \,d^{3} e^{4} x^{3}-40 c^{2} d^{4} e^{3} x^{3}+19305 a^{2} d \,e^{6} x^{2}+21450 a b \,d^{2} e^{5} x^{2}+390 a c \,d^{3} e^{4} x^{2}+195 b^{2} d^{3} e^{4} x^{2}-180 b c \,d^{4} e^{3} x^{2}+48 c^{2} d^{5} e^{2} x^{2}+19305 a^{2} d^{2} e^{5} x +1430 a b \,d^{3} e^{4} x -520 a c \,d^{4} e^{3} x -260 b^{2} d^{4} e^{3} x +240 b c \,d^{5} e^{2} x -64 c^{2} d^{6} e x +6435 a^{2} d^{3} e^{4}-2860 a b \,d^{4} e^{3}+1040 a c \,d^{5} e^{2}+520 b^{2} d^{5} e^{2}-480 b c \,d^{6} e +128 c^{2} d^{7}\right ) \sqrt {e x +d}}{45045 e^{5}}\)	\(433\)

[In]

int((e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/15*c^2*(e*x+d)^(15/2)+2/13*c*(b*e-2*c*d)*(e*x+d)^(13/2)+1/11*(2*(a*e^2-b*d*e+c*d^2)*c+(b*e-2*c*d)^2)*
(e*x+d)^(11/2)+2/9*(a*e^2-b*d*e+c*d^2)*(b*e-2*c*d)*(e*x+d)^(9/2)+1/7*(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(7/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (146) = 292\).

Time = 0.33 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.20 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, c^{2} e^{7} x^{7} + 128 \, c^{2} d^{7} - 480 \, b c d^{6} e - 2860 \, a b d^{4} e^{3} + 6435 \, a^{2} d^{3} e^{4} + 520 \, {\left (b^{2} + 2 \, a c\right )} d^{5} e^{2} + 231 \, {\left (31 \, c^{2} d e^{6} + 30 \, b c e^{7}\right )} x^{6} + 63 \, {\left (71 \, c^{2} d^{2} e^{5} + 270 \, b c d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} e^{7}\right )} x^{5} + 35 \, {\left (c^{2} d^{3} e^{4} + 318 \, b c d^{2} e^{5} + 286 \, a b e^{7} + 299 \, {\left (b^{2} + 2 \, a c\right )} d e^{6}\right )} x^{4} - 5 \, {\left (8 \, c^{2} d^{4} e^{3} - 30 \, b c d^{3} e^{4} - 5434 \, a b d e^{6} - 1287 \, a^{2} e^{7} - 1469 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{5}\right )} x^{3} + 3 \, {\left (16 \, c^{2} d^{5} e^{2} - 60 \, b c d^{4} e^{3} + 7150 \, a b d^{2} e^{5} + 6435 \, a^{2} d e^{6} + 65 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{2} - {\left (64 \, c^{2} d^{6} e - 240 \, b c d^{5} e^{2} - 1430 \, a b d^{3} e^{4} - 19305 \, a^{2} d^{2} e^{5} + 260 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*c^2*e^7*x^7 + 128*c^2*d^7 - 480*b*c*d^6*e - 2860*a*b*d^4*e^3 + 6435*a^2*d^3*e^4 + 520*(b^2 + 2*a
*c)*d^5*e^2 + 231*(31*c^2*d*e^6 + 30*b*c*e^7)*x^6 + 63*(71*c^2*d^2*e^5 + 270*b*c*d*e^6 + 65*(b^2 + 2*a*c)*e^7)
*x^5 + 35*(c^2*d^3*e^4 + 318*b*c*d^2*e^5 + 286*a*b*e^7 + 299*(b^2 + 2*a*c)*d*e^6)*x^4 - 5*(8*c^2*d^4*e^3 - 30*
b*c*d^3*e^4 - 5434*a*b*d*e^6 - 1287*a^2*e^7 - 1469*(b^2 + 2*a*c)*d^2*e^5)*x^3 + 3*(16*c^2*d^5*e^2 - 60*b*c*d^4
*e^3 + 7150*a*b*d^2*e^5 + 6435*a^2*d*e^6 + 65*(b^2 + 2*a*c)*d^3*e^4)*x^2 - (64*c^2*d^6*e - 240*b*c*d^5*e^2 - 1
430*a*b*d^3*e^4 - 19305*a^2*d^2*e^5 + 260*(b^2 + 2*a*c)*d^4*e^3)*x)*sqrt(e*x + d)/e^5

Sympy [A] (verification not implemented)

Time = 1.08 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.67 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{4}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{13 e^{4}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{7 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**(5/2)*(c*x**2+b*x+a)**2,x)

[Out]

Piecewise((2*(c**2*(d + e*x)**(15/2)/(15*e**4) + (d + e*x)**(13/2)*(2*b*c*e - 4*c**2*d)/(13*e**4) + (d + e*x)*
*(11/2)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)/(11*e**4) + (d + e*x)**(9/2)*(2*a*b*e**3 - 4*a*c*d*
e**2 - 2*b**2*d*e**2 + 6*b*c*d**2*e - 4*c**2*d**3)/(9*e**4) + (d + e*x)**(7/2)*(a**2*e**4 - 2*a*b*d*e**3 + 2*a
*c*d**2*e**2 + b**2*d**2*e**2 - 2*b*c*d**3*e + c**2*d**4)/(7*e**4))/e, Ne(e, 0)), (d**(5/2)*(a**2*x + a*b*x**2
+ b*c*x**4/2 + c**2*x**5/5 + x**3*(2*a*c + b**2)/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.06 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{\frac {15}{2}} c^{2} - 6930 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 4095 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 10010 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 6435 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{45045 \, e^{5}} \]

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 6930*(2*c^2*d - b*c*e)*(e*x + d)^(13/2) + 4095*(6*c^2*d^2 - 6*b*c*d*e + (
b^2 + 2*a*c)*e^2)*(e*x + d)^(11/2) - 10010*(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*(e*x + d)
^(9/2) + 6435*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*(e*x + d)^(7/2))/e^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1411 vs. \(2 (146) = 292\).

Time = 0.29 (sec) , antiderivative size = 1411, normalized size of antiderivative = 8.50 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(5/2)*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(e*x + d)*a^2*d^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^2 + 30030*((e*x + d)^
(3/2) - 3*sqrt(e*x + d)*d)*a*b*d^3/e + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*
a^2*d + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2*d^3/e^2 + 6006*(3*(e*x + d)
^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c*d^3/e^2 + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3
/2)*d + 15*sqrt(e*x + d)*d^2)*a*b*d^2/e + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*
d^2 - 35*sqrt(e*x + d)*d^3)*a^2 + 2574*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*b*c*d^3/e^3 + 3861*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*b^2*d^2/e^2 + 7722*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*a*c*d^2/e^2 + 7722*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35
*sqrt(e*x + d)*d^3)*a*b*d/e + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*
(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^3/e^4 + 858*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 3
78*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b*c*d^2/e^3 + 429*(35*(e*x + d)^(9/2
) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^2*d/e
^2 + 858*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315
*sqrt(e*x + d)*d^4)*a*c*d/e^2 + 286*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 42
0*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b/e + 195*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*
(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*c^2*d^2/e^4
+ 390*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 115
5*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b*c*d/e^3 + 65*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 9
90*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^2/e^2
+ 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155
*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*a*c/e^2 + 45*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5
005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 300
3*sqrt(e*x + d)*d^6)*c^2*d/e^4 + 30*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2
- 8580*(e*x + d)^(7/2)*d^3 + 9009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)*d^6)*b*
c/e^3 + 7*(429*(e*x + d)^(15/2) - 3465*(e*x + d)^(13/2)*d + 12285*(e*x + d)^(11/2)*d^2 - 25025*(e*x + d)^(9/2)
*d^3 + 32175*(e*x + d)^(7/2)*d^4 - 27027*(e*x + d)^(5/2)*d^5 + 15015*(e*x + d)^(3/2)*d^6 - 6435*sqrt(e*x + d)*
d^7)*c^2/e^4)/e

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.89 \[ \int (d+e x)^{5/2} \left (a+b x+c x^2\right )^2 \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{15/2}}{15\,e^5}+\frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{11\,e^5}+\frac {2\,{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d^2-b\,d\,e+a\,e^2\right )}^2}{7\,e^5}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{9\,e^5} \]

[In]

int((d + e*x)^(5/2)*(a + b*x + c*x^2)^2,x)

[Out]

(2*c^2*(d + e*x)^(15/2))/(15*e^5) + ((d + e*x)^(11/2)*(2*b^2*e^2 + 12*c^2*d^2 + 4*a*c*e^2 - 12*b*c*d*e))/(11*e
^5) + (2*(d + e*x)^(7/2)*(a*e^2 + c*d^2 - b*d*e)^2)/(7*e^5) - ((8*c^2*d - 4*b*c*e)*(d + e*x)^(13/2))/(13*e^5)
+ (4*(b*e - 2*c*d)*(d + e*x)^(9/2)*(a*e^2 + c*d^2 - b*d*e))/(9*e^5)