\(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2) \, dx\) [1621]

Optimal result

Integrand size = 26, antiderivative size = 71 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (b d-a e)^2 (d+e x)^{9/2}}{9 e^3}-\frac {4 b (b d-a e) (d+e x)^{11/2}}{11 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \]

[Out]

2/9*(-a*e+b*d)^2*(e*x+d)^(9/2)/e^3-4/11*b*(-a*e+b*d)*(e*x+d)^(11/2)/e^3+2/13*b^2*(e*x+d)^(13/2)/e^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac {2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac {2 b^2 (d+e x)^{13/2}}{13 e^3} \]

[In]

Int[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(9/2))/(9*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^3) + (2*b^2*(d + e*x)^(13
/2))/(13*e^3)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (-2 d+9 e x)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

[In]

Integrate[(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(2*(d + e*x)^(9/2)*(143*a^2*e^2 + 26*a*b*e*(-2*d + 9*e*x) + b^2*(8*d^2 - 36*d*e*x + 99*e^2*x^2)))/(1287*e^3)

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76

method	result	size
pseudoelliptic	\(\frac {2 \left (\left (\frac {9}{13} b^{2} x^{2}+\frac {18}{11} a b x +a^{2}\right ) e^{2}-\frac {4 \left (\frac {9 b x}{13}+a \right ) b d e}{11}+\frac {8 b^{2} d^{2}}{143}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{3}}\)	\(54\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (99 x^{2} b^{2} e^{2}+234 x a b \,e^{2}-36 b^{2} d e x +143 a^{2} e^{2}-52 a b d e +8 b^{2} d^{2}\right )}{1287 e^{3}}\)	\(63\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\)	\(70\)
default	\(\frac {\frac {2 b^{2} \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{3}}\)	\(70\)
trager	\(\frac {2 \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 a^{2} e^{6} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\)	\(223\)
risch	\(\frac {2 \left (99 b^{2} e^{6} x^{6}+234 a b \,e^{6} x^{5}+360 b^{2} d \,e^{5} x^{5}+143 a^{2} e^{6} x^{4}+884 a b d \,e^{5} x^{4}+458 b^{2} d^{2} e^{4} x^{4}+572 a^{2} d \,e^{5} x^{3}+1196 a b \,d^{2} e^{4} x^{3}+212 b^{2} d^{3} e^{3} x^{3}+858 a^{2} d^{2} e^{4} x^{2}+624 a b \,d^{3} e^{3} x^{2}+3 b^{2} d^{4} e^{2} x^{2}+572 a^{2} d^{3} e^{3} x +26 a b \,d^{4} e^{2} x -4 b^{2} d^{5} e x +143 a^{2} d^{4} e^{2}-52 a b \,d^{5} e +8 b^{2} d^{6}\right ) \sqrt {e x +d}}{1287 e^{3}}\)	\(223\)

[In]

int((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x,method=_RETURNVERBOSE)

[Out]

2/9*((9/13*b^2*x^2+18/11*a*b*x+a^2)*e^2-4/11*(9/13*b*x+a)*b*d*e+8/143*b^2*d^2)*(e*x+d)^(9/2)/e^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (59) = 118\).

Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.99 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (99 \, b^{2} e^{6} x^{6} + 8 \, b^{2} d^{6} - 52 \, a b d^{5} e + 143 \, a^{2} d^{4} e^{2} + 18 \, {\left (20 \, b^{2} d e^{5} + 13 \, a b e^{6}\right )} x^{5} + {\left (458 \, b^{2} d^{2} e^{4} + 884 \, a b d e^{5} + 143 \, a^{2} e^{6}\right )} x^{4} + 4 \, {\left (53 \, b^{2} d^{3} e^{3} + 299 \, a b d^{2} e^{4} + 143 \, a^{2} d e^{5}\right )} x^{3} + 3 \, {\left (b^{2} d^{4} e^{2} + 208 \, a b d^{3} e^{3} + 286 \, a^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (2 \, b^{2} d^{5} e - 13 \, a b d^{4} e^{2} - 286 \, a^{2} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{1287 \, e^{3}} \]

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="fricas")

[Out]

2/1287*(99*b^2*e^6*x^6 + 8*b^2*d^6 - 52*a*b*d^5*e + 143*a^2*d^4*e^2 + 18*(20*b^2*d*e^5 + 13*a*b*e^6)*x^5 + (45
8*b^2*d^2*e^4 + 884*a*b*d*e^5 + 143*a^2*e^6)*x^4 + 4*(53*b^2*d^3*e^3 + 299*a*b*d^2*e^4 + 143*a^2*d*e^5)*x^3 +
3*(b^2*d^4*e^2 + 208*a*b*d^3*e^3 + 286*a^2*d^2*e^4)*x^2 - 2*(2*b^2*d^5*e - 13*a*b*d^4*e^2 - 286*a^2*d^3*e^3)*x
)*sqrt(e*x + d)/e^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (65) = 130\).

Time = 0.56 (sec) , antiderivative size = 432, normalized size of antiderivative = 6.08 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\begin {cases} \frac {2 a^{2} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{2} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{2} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{2} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{2} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {8 a b d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {4 a b d^{4} x \sqrt {d + e x}}{99 e} + \frac {32 a b d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {184 a b d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {136 a b d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {4 a b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 b^{2} d^{6} \sqrt {d + e x}}{1287 e^{3}} - \frac {8 b^{2} d^{5} x \sqrt {d + e x}}{1287 e^{2}} + \frac {2 b^{2} d^{4} x^{2} \sqrt {d + e x}}{429 e} + \frac {424 b^{2} d^{3} x^{3} \sqrt {d + e x}}{1287} + \frac {916 b^{2} d^{2} e x^{4} \sqrt {d + e x}}{1287} + \frac {80 b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {2 b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**2*d**4*sqrt(d + e*x)/(9*e) + 8*a**2*d**3*x*sqrt(d + e*x)/9 + 4*a**2*d**2*e*x**2*sqrt(d + e*x)/
3 + 8*a**2*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**2*e**3*x**4*sqrt(d + e*x)/9 - 8*a*b*d**5*sqrt(d + e*x)/(99*e**2)
+ 4*a*b*d**4*x*sqrt(d + e*x)/(99*e) + 32*a*b*d**3*x**2*sqrt(d + e*x)/33 + 184*a*b*d**2*e*x**3*sqrt(d + e*x)/9
9 + 136*a*b*d*e**2*x**4*sqrt(d + e*x)/99 + 4*a*b*e**3*x**5*sqrt(d + e*x)/11 + 16*b**2*d**6*sqrt(d + e*x)/(1287
*e**3) - 8*b**2*d**5*x*sqrt(d + e*x)/(1287*e**2) + 2*b**2*d**4*x**2*sqrt(d + e*x)/(429*e) + 424*b**2*d**3*x**3
*sqrt(d + e*x)/1287 + 916*b**2*d**2*e*x**4*sqrt(d + e*x)/1287 + 80*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 2*b**2
*e**3*x**6*sqrt(d + e*x)/13, Ne(e, 0)), (d**(7/2)*(a**2*x + a*b*x**2 + b**2*x**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (99 \, {\left (e x + d\right )}^{\frac {13}{2}} b^{2} - 234 \, {\left (b^{2} d - a b e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 143 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{1287 \, e^{3}} \]

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="maxima")

[Out]

2/1287*(99*(e*x + d)^(13/2)*b^2 - 234*(b^2*d - a*b*e)*(e*x + d)^(11/2) + 143*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(
e*x + d)^(9/2))/e^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (59) = 118\).

Time = 0.28 (sec) , antiderivative size = 791, normalized size of antiderivative = 11.14 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2),x, algorithm="giac")

[Out]

2/45045*(45045*sqrt(e*x + d)*a^2*d^4 + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^3 + 30030*((e*x + d)^
(3/2) - 3*sqrt(e*x + d)*d)*a*b*d^4/e + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)
*a^2*d^2 + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*b^2*d^4/e^2 + 24024*(3*(e*x
+ d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b*d^3/e + 5148*(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*d + 5148*(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^3/e^2 + 15444*(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^2/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)*a^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)*b^2*d^2/e^2
+ 1144*(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*b*d/e + 260*(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)*b^2*d/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*b/e + 15*(231*(e*x + d)^(13/2) - 1638*(e*x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 858
0*(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^2/e^2)
/e

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2\,{\left (d+e\,x\right )}^{9/2}\,\left (99\,b^2\,{\left (d+e\,x\right )}^2+143\,a^2\,e^2+143\,b^2\,d^2-234\,b^2\,d\,\left (d+e\,x\right )+234\,a\,b\,e\,\left (d+e\,x\right )-286\,a\,b\,d\,e\right )}{1287\,e^3} \]

[In]

int((d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x),x)

[Out]

(2*(d + e*x)^(9/2)*(99*b^2*(d + e*x)^2 + 143*a^2*e^2 + 143*b^2*d^2 - 234*b^2*d*(d + e*x) + 234*a*b*e*(d + e*x)
- 286*a*b*d*e))/(1287*e^3)