### 3.1624 $$\int \sqrt{d+e x} (a^2+2 a b x+b^2 x^2) \, dx$$

Optimal. Leaf size=71 $-\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3}$

[Out]

(2*(b*d - a*e)^2*(d + e*x)^(3/2))/(3*e^3) - (4*b*(b*d - a*e)*(d + e*x)^(5/2))/(5*e^3) + (2*b^2*(d + e*x)^(7/2)
)/(7*e^3)

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Rubi [A]  time = 0.0225009, 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} $-\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0343695, size = 61, normalized size = 0.86 $\frac{2 (d+e x)^{3/2} \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(35*a^2*e^2 + 14*a*b*e*(-2*d + 3*e*x) + b^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)

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Maple [A]  time = 0.046, size = 63, normalized size = 0.9 \begin{align*}{\frac{30\,{b}^{2}{x}^{2}{e}^{2}+84\,xab{e}^{2}-24\,x{b}^{2}de+70\,{a}^{2}{e}^{2}-56\,abde+16\,{b}^{2}{d}^{2}}{105\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x)

[Out]

2/105*(e*x+d)^(3/2)*(15*b^2*e^2*x^2+42*a*b*e^2*x-12*b^2*d*e*x+35*a^2*e^2-28*a*b*d*e+8*b^2*d^2)/e^3

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Maxima [A]  time = 1.08315, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{2} - 42 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*b^2 - 42*(b^2*d - a*b*e)*(e*x + d)^(5/2) + 35*(b^2*d^2 - 2*a*b*d*e + a^2*e^2)*(e*x +
d)^(3/2))/e^3

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Fricas [A]  time = 1.56796, size = 220, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (15 \, b^{2} e^{3} x^{3} + 8 \, b^{2} d^{3} - 28 \, a b d^{2} e + 35 \, a^{2} d e^{2} + 3 \,{\left (b^{2} d e^{2} + 14 \, a b e^{3}\right )} x^{2} -{\left (4 \, b^{2} d^{2} e - 14 \, a b d e^{2} - 35 \, a^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/105*(15*b^2*e^3*x^3 + 8*b^2*d^3 - 28*a*b*d^2*e + 35*a^2*d*e^2 + 3*(b^2*d*e^2 + 14*a*b*e^3)*x^2 - (4*b^2*d^2*
e - 14*a*b*d*e^2 - 35*a^2*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 2.83358, size = 85, normalized size = 1.2 \begin{align*} \frac{2 \left (\frac{b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 a b e - 2 b^{2} d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} e^{2} - 2 a b d e + b^{2} d^{2}\right )}{3 e^{2}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**2+2*a*b*x+a**2)*(e*x+d)**(1/2),x)

[Out]

2*(b**2*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(5/2)*(2*a*b*e - 2*b**2*d)/(5*e**2) + (d + e*x)**(3/2)*(a**2*e*
*2 - 2*a*b*d*e + b**2*d**2)/(3*e**2))/e

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Giac [A]  time = 1.18315, size = 117, normalized size = 1.65 \begin{align*} \frac{2}{105} \,{\left (14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b e^{\left (-1\right )} +{\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} e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^2+2*a*b*x+a^2)*(e*x+d)^(1/2),x, algorithm="giac")

[Out]

2/105*(14*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*b*e^(-1) + (15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d +
35*(x*e + d)^(3/2)*d^2)*b^2*e^(-2) + 35*(x*e + d)^(3/2)*a^2)*e^(-1)