### 3.1976 $$\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2) \, dx$$

Optimal. Leaf size=43 $\frac{2}{7} (d+e x)^{7/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{9/2}}{9 e^2}$

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(7/2))/7 + (2*c*d*(d + e*x)^(9/2))/(9*e^2)

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Rubi [A]  time = 0.0199979, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 43} $\frac{2}{7} (d+e x)^{7/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{9/2}}{9 e^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*(a - (c*d^2)/e^2)*(d + e*x)^(7/2))/7 + (2*c*d*(d + e*x)^(9/2))/(9*e^2)

Rule 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 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)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^{5/2}}{e}+\frac{c d (d+e x)^{7/2}}{e}\right ) \, dx\\ &=\frac{2}{7} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^{7/2}+\frac{2 c d (d+e x)^{9/2}}{9 e^2}\\ \end{align*}

Mathematica [A]  time = 0.0337937, size = 34, normalized size = 0.79 $\frac{2 (d+e x)^{7/2} \left (9 a e^2+c d (7 e x-2 d)\right )}{63 e^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(2*(d + e*x)^(7/2)*(9*a*e^2 + c*d*(-2*d + 7*e*x)))/(63*e^2)

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Maple [A]  time = 0.042, size = 32, normalized size = 0.7 \begin{align*}{\frac{14\,cdex+18\,a{e}^{2}-4\,c{d}^{2}}{63\,{e}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

2/63*(e*x+d)^(7/2)*(7*c*d*e*x+9*a*e^2-2*c*d^2)/e^2

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Maxima [A]  time = 0.985135, size = 51, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (7 \,{\left (e x + d\right )}^{\frac{9}{2}} c d - 9 \,{\left (c d^{2} - a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{63 \, e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="maxima")

[Out]

2/63*(7*(e*x + d)^(9/2)*c*d - 9*(c*d^2 - a*e^2)*(e*x + d)^(7/2))/e^2

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Fricas [B]  time = 2.33905, size = 213, normalized size = 4.95 \begin{align*} \frac{2 \,{\left (7 \, c d e^{4} x^{4} - 2 \, c d^{5} + 9 \, a d^{3} e^{2} +{\left (19 \, c d^{2} e^{3} + 9 \, a e^{5}\right )} x^{3} + 3 \,{\left (5 \, c d^{3} e^{2} + 9 \, a d e^{4}\right )} x^{2} +{\left (c d^{4} e + 27 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{63 \, e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="fricas")

[Out]

2/63*(7*c*d*e^4*x^4 - 2*c*d^5 + 9*a*d^3*e^2 + (19*c*d^2*e^3 + 9*a*e^5)*x^3 + 3*(5*c*d^3*e^2 + 9*a*d*e^4)*x^2 +
(c*d^4*e + 27*a*d^2*e^3)*x)*sqrt(e*x + d)/e^2

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Sympy [A]  time = 12.7528, size = 235, normalized size = 5.47 \begin{align*} a d^{2} e \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + 4 a d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 2 a \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right ) + \frac{2 c d^{3} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 c d^{2} \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(3/2)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

a*d**2*e*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a*d*(-d*(d + e*x)**(3/2)/3 + (
d + e*x)**(5/2)/5) + 2*a*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7) + 2*c*d**3*(-
d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 4*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 +
(d + e*x)**(7/2)/7)/e**2 + 2*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)
/7 + (d + e*x)**(9/2)/9)/e**2

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Giac [B]  time = 1.13888, size = 290, normalized size = 6.74 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} c d^{3} e^{\left (-1\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2} e + 6 \,{\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 )} c d^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d e +{\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 )} c d e^{\left (-1\right )} + 3 \,{\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 e\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*c*d^3*e^(-1) + 105*(x*e + d)^(3/2)*a*d^2*e + 6*(15*(x*e +
d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*c*d^2*e^(-1) + 42*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(
3/2)*d)*a*d*e + (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)*c*d*e^(-1) + 3*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*e)*e^(-1)