3.1622 \(\int (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2) \, dx\)

Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{9/2} (b d-a e)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e)^2}{7 e^3}+\frac{2 b^2 (d+e x)^{11/2}}{11 e^3} \]

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^3) + (2*b^2*(d + e*x)^(11/2
))/(11*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0233756, 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} \[ -\frac{4 b (d+e x)^{9/2} (b d-a e)}{9 e^3}+\frac{2 (d+e x)^{7/2} (b d-a e)^2}{7 e^3}+\frac{2 b^2 (d+e x)^{11/2}}{11 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(7/2))/(7*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(9/2))/(9*e^3) + (2*b^2*(d + e*x)^(11/2
))/(11*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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (d+e x)^{5/2}}{e^2}-\frac{2 b (b d-a e) (d+e x)^{7/2}}{e^2}+\frac{b^2 (d+e x)^{9/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^3}-\frac{4 b (b d-a e) (d+e x)^{9/2}}{9 e^3}+\frac{2 b^2 (d+e x)^{11/2}}{11 e^3}\\ \end{align*}

Mathematica [A]  time = 0.0413262, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{7/2} \left (99 a^2 e^2+22 a b e (7 e x-2 d)+b^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(99*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)))/(693*e^3)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 63, normalized size = 0.9 \begin{align*}{\frac{126\,{b}^{2}{x}^{2}{e}^{2}+308\,xab{e}^{2}-56\,x{b}^{2}de+198\,{a}^{2}{e}^{2}-88\,abde+16\,{b}^{2}{d}^{2}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(e*x+d)^(7/2)*(63*b^2*e^2*x^2+154*a*b*e^2*x-28*b^2*d*e*x+99*a^2*e^2-44*a*b*d*e+8*b^2*d^2)/e^3

________________________________________________________________________________________

Maxima [A]  time = 1.06185, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} b^{2} - 154 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*(e*x + d)^(11/2)*b^2 - 154*(b^2*d - a*b*e)*(e*x + d)^(9/2) + 99*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x
+ d)^(7/2))/e^3

________________________________________________________________________________________

Fricas [B]  time = 1.53275, size = 382, normalized size = 5.38 \begin{align*} \frac{2 \,{\left (63 \, b^{2} e^{5} x^{5} + 8 \, b^{2} d^{5} - 44 \, a b d^{4} e + 99 \, a^{2} d^{3} e^{2} + 7 \,{\left (23 \, b^{2} d e^{4} + 22 \, a b e^{5}\right )} x^{4} +{\left (113 \, b^{2} d^{2} e^{3} + 418 \, a b d e^{4} + 99 \, a^{2} e^{5}\right )} x^{3} + 3 \,{\left (b^{2} d^{3} e^{2} + 110 \, a b d^{2} e^{3} + 99 \, a^{2} d e^{4}\right )} x^{2} -{\left (4 \, b^{2} d^{4} e - 22 \, a b d^{3} e^{2} - 297 \, a^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*b^2*e^5*x^5 + 8*b^2*d^5 - 44*a*b*d^4*e + 99*a^2*d^3*e^2 + 7*(23*b^2*d*e^4 + 22*a*b*e^5)*x^4 + (113*b
^2*d^2*e^3 + 418*a*b*d*e^4 + 99*a^2*e^5)*x^3 + 3*(b^2*d^3*e^2 + 110*a*b*d^2*e^3 + 99*a^2*d*e^4)*x^2 - (4*b^2*d
^4*e - 22*a*b*d^3*e^2 - 297*a^2*d^2*e^3)*x)*sqrt(e*x + d)/e^3

________________________________________________________________________________________

Sympy [A]  time = 3.87907, size = 355, normalized size = 5. \begin{align*} \begin{cases} \frac{2 a^{2} d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a^{2} d^{2} x \sqrt{d + e x}}{7} + \frac{6 a^{2} d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a^{2} e^{2} x^{3} \sqrt{d + e x}}{7} - \frac{8 a b d^{4} \sqrt{d + e x}}{63 e^{2}} + \frac{4 a b d^{3} x \sqrt{d + e x}}{63 e} + \frac{20 a b d^{2} x^{2} \sqrt{d + e x}}{21} + \frac{76 a b d e x^{3} \sqrt{d + e x}}{63} + \frac{4 a b e^{2} x^{4} \sqrt{d + e x}}{9} + \frac{16 b^{2} d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 b^{2} d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 b^{2} d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 b^{2} d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 b^{2} d e x^{4} \sqrt{d + e x}}{99} + \frac{2 b^{2} e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{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)**(5/2)*(b**2*x**2+2*a*b*x+a**2),x)

[Out]

Piecewise((2*a**2*d**3*sqrt(d + e*x)/(7*e) + 6*a**2*d**2*x*sqrt(d + e*x)/7 + 6*a**2*d*e*x**2*sqrt(d + e*x)/7 +
2*a**2*e**2*x**3*sqrt(d + e*x)/7 - 8*a*b*d**4*sqrt(d + e*x)/(63*e**2) + 4*a*b*d**3*x*sqrt(d + e*x)/(63*e) + 2
0*a*b*d**2*x**2*sqrt(d + e*x)/21 + 76*a*b*d*e*x**3*sqrt(d + e*x)/63 + 4*a*b*e**2*x**4*sqrt(d + e*x)/9 + 16*b**
2*d**5*sqrt(d + e*x)/(693*e**3) - 8*b**2*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*b**2*d**3*x**2*sqrt(d + e*x)/(231
*e) + 226*b**2*d**2*x**3*sqrt(d + e*x)/693 + 46*b**2*d*e*x**4*sqrt(d + e*x)/99 + 2*b**2*e**2*x**5*sqrt(d + e*x
)/11, Ne(e, 0)), (d**(5/2)*(a**2*x + a*b*x**2 + b**2*x**3/3), True))

________________________________________________________________________________________

Giac [B]  time = 1.16392, size = 518, normalized size = 7.3 \begin{align*} \frac{2}{3465} \,{\left (462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b d^{2} e^{\left (-1\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{2} d^{2} e^{\left (-2\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d^{2} + 132 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a b d e^{\left (-1\right )} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b^{2} d e^{\left (-2\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} d + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a b e^{\left (-1\right )} +{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} b^{2} e^{\left (-2\right )} + 33 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a^{2}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*d^2*e^(-1) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(
5/2)*d + 35*(x*e + d)^(3/2)*d^2)*b^2*d^2*e^(-2) + 1155*(x*e + d)^(3/2)*a^2*d^2 + 132*(15*(x*e + d)^(7/2) - 42*
(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*b*d*e^(-1) + 22*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 18
9*(x*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*b^2*d*e^(-2) + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*
a^2*d + 22*(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*e^(-1) + (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*e^(-2) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*
d^2)*a^2)*e^(-1)