3.1984 \(\int \sqrt{d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\)

Optimal. Leaf size=83 \[ -\frac{4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0577019, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {626, 43} \[ -\frac{4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac{2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0557433, size = 67, normalized size = 0.81 \[ \frac{2 (d+e x)^{7/2} \left (99 a^2 e^4-22 a c d e^2 (2 d-7 e x)+c^2 d^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(99*a^2*e^4 - 22*a*c*d*e^2*(2*d - 7*e*x) + c^2*d^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e
^3)

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Maple [A]  time = 0.046, size = 73, normalized size = 0.9 \begin{align*}{\frac{126\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+308\,acd{e}^{3}x-56\,{c}^{2}{d}^{3}ex+198\,{a}^{2}{e}^{4}-88\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{693\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(e*x+d)^(7/2)*(63*c^2*d^2*e^2*x^2+154*a*c*d*e^3*x-28*c^2*d^3*e*x+99*a^2*e^4-44*a*c*d^2*e^2+8*c^2*d^4)/e^
3

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Maxima [A]  time = 0.98391, size = 108, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} d^{2} - 154 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 99 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/693*(63*(e*x + d)^(11/2)*c^2*d^2 - 154*(c^2*d^3 - a*c*d*e^2)*(e*x + d)^(9/2) + 99*(c^2*d^4 - 2*a*c*d^2*e^2 +
a^2*e^4)*(e*x + d)^(7/2))/e^3

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Fricas [B]  time = 1.92318, size = 398, normalized size = 4.8 \begin{align*} \frac{2 \,{\left (63 \, c^{2} d^{2} e^{5} x^{5} + 8 \, c^{2} d^{7} - 44 \, a c d^{5} e^{2} + 99 \, a^{2} d^{3} e^{4} + 7 \,{\left (23 \, c^{2} d^{3} e^{4} + 22 \, a c d e^{6}\right )} x^{4} +{\left (113 \, c^{2} d^{4} e^{3} + 418 \, a c d^{2} e^{5} + 99 \, a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} + 110 \, a c d^{3} e^{4} + 99 \, a^{2} d e^{6}\right )} x^{2} -{\left (4 \, c^{2} d^{6} e - 22 \, a c d^{4} e^{3} - 297 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*c^2*d^2*e^5*x^5 + 8*c^2*d^7 - 44*a*c*d^5*e^2 + 99*a^2*d^3*e^4 + 7*(23*c^2*d^3*e^4 + 22*a*c*d*e^6)*x^
4 + (113*c^2*d^4*e^3 + 418*a*c*d^2*e^5 + 99*a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 + 110*a*c*d^3*e^4 + 99*a^2*d*e^6)*x^
2 - (4*c^2*d^6*e - 22*a*c*d^4*e^3 - 297*a^2*d^2*e^5)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.82981, size = 97, normalized size = 1.17 \begin{align*} \frac{2 \left (\frac{c^{2} d^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{2}} + \frac{\left (d + e x\right )^{\frac{9}{2}} \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{9 e^{2}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{7 e^{2}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.16618, size = 529, normalized size = 6.37 \begin{align*} \frac{2}{3465} \,{\left (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 )} c^{2} d^{4} e^{\left (-2\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 )} c^{2} d^{3} e^{\left (-2\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d^{2} e^{2} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a c d^{3} +{\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 )} c^{2} d^{2} e^{\left (-2\right )} + 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 c d^{2} + 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 e^{2} + 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 c d + 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} e^{2}\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*c^2*d^4*e^(-2) + 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)*c^2*d^3*e^(-2) + 1155*(
x*e + d)^(3/2)*a^2*d^2*e^2 + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*c*d^3 + (315*(x*e + d)^(11/2) - 1
540*(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)*c^2*d^
2*e^(-2) + 132*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c*d^2 + 462*(3*(x*e + d)
^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*d*e^2 + 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*c*d + 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^2)*e^(-1)