3.1621 \(\int (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac{2 b^2 (d+e x)^{13/2}}{13 e^3} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.0283148, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ -\frac{4 b (d+e x)^{11/2} (b d-a e)}{11 e^3}+\frac{2 (d+e x)^{9/2} (b d-a e)^2}{9 e^3}+\frac{2 b^2 (d+e x)^{13/2}}{13 e^3} \]

Antiderivative was successfully verified.

[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 43

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*} \int (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\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]  time = 0.0532309, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{9/2} \left (143 a^2 e^2+26 a b e (9 e x-2 d)+b^2 \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )}{1287 e^3} \]

Antiderivative was successfully verified.

[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]  time = 0.046, size = 63, normalized size = 0.9 \begin{align*}{\frac{198\,{b}^{2}{x}^{2}{e}^{2}+468\,xab{e}^{2}-72\,x{b}^{2}de+286\,{a}^{2}{e}^{2}-104\,abde+16\,{b}^{2}{d}^{2}}{1287\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1287*(e*x+d)^(9/2)*(99*b^2*e^2*x^2+234*a*b*e^2*x-36*b^2*d*e*x+143*a^2*e^2-52*a*b*d*e+8*b^2*d^2)/e^3

________________________________________________________________________________________

Maxima [A]  time = 1.21825, size = 92, normalized size = 1.3 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [B]  time = 1.52034, size = 474, normalized size = 6.68 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 9.58377, size = 432, normalized size = 6.08 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [B]  time = 1.16192, size = 803, normalized size = 11.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d^3*e^(-1) + 429*(15*(x*e + d)^(7/2) - 42*(x*e + d
)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^3*e^(-2) + 15015*(x*e + d)^(3/2)*a^2*d^3 + 2574*(15*(x*e + d)^(7/2)
- 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*d^2*e^(-1) + 429*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2
)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^2*d^2*e^(-2) + 9009*(3*(x*e + d)^(5/2) - 5*(x*e + d
)^(3/2)*d)*a^2*d^2 + 858*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)
^(3/2)*d^3)*a*b*d*e^(-1) + 39*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772
*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*b^2*d*e^(-2) + 1287*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*
d + 35*(x*e + d)^(3/2)*d^2)*a^2*d + 26*(315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d
^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*a*b*e^(-1) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d
)^(11/2)*d + 10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)
^(3/2)*d^5)*b^2*e^(-2) + 143*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x*e + d)^(5/2)*d^2 - 105*(x*e
+ d)^(3/2)*d^3)*a^2)*e^(-1)