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

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

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Rubi [A]  time = 0.06, antiderivative size = 83, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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} \end {gather*}

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 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])

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]

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*}

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Mathematica [A]  time = 0.06, size = 67, normalized size = 0.81 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 84, normalized size = 1.01 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (99 a^2 e^4-198 a c d^2 e^2+154 a c d e^2 (d+e x)+99 c^2 d^4-154 c^2 d^3 (d+e x)+63 c^2 d^2 (d+e x)^2\right )}{693 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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*c^2*d^4 - 198*a*c*d^2*e^2 + 99*a^2*e^4 - 154*c^2*d^3*(d + e*x) + 154*a*c*d*e^2*(d + e*x
) + 63*c^2*d^2*(d + e*x)^2))/(693*e^3)

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fricas [B]  time = 0.40, size = 184, normalized size = 2.22 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.20, size = 599, normalized size = 7.22 \begin {gather*} \frac {2}{3465} \, {\left (231 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} c^{2} d^{5} e^{\left (-2\right )} + 297 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} c^{2} d^{4} e^{\left (-2\right )} + 2310 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a c d^{4} + 33 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} c^{2} d^{3} e^{\left (-2\right )} + 3465 \, \sqrt {x e + d} a^{2} d^{3} e^{2} + 1386 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a c d^{3} + 3465 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} a^{2} d^{2} e^{2} + 5 \, {\left (63 \, {\left (x e + d\right )}^{\frac {11}{2}} - 385 \, {\left (x e + d\right )}^{\frac {9}{2}} d + 990 \, {\left (x e + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {x e + d} d^{5}\right )} c^{2} d^{2} e^{\left (-2\right )} + 594 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a c d^{2} + 693 \, {\left (3 \, {\left (x e + d\right )}^{\frac {5}{2}} - 10 \, {\left (x e + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {x e + d} d^{2}\right )} a^{2} d e^{2} + 22 \, {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} a c d + 99 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} a^{2} e^{2}\right )} e^{\left (-1\right )} \end {gather*}

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*(231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c^2*d^5*e^(-2) + 297*(5*(x*e + d
)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*c^2*d^4*e^(-2) + 2310*((x*e +
d)^(3/2) - 3*sqrt(x*e + d)*d)*a*c*d^4 + 33*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d
^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*d^3*e^(-2) + 3465*sqrt(x*e + d)*a^2*d^3*e^2 + 1386*(
3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*d^3 + 3465*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*a^2*d^2*e^2 + 5*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d
)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*c^2*d^2*e^(-2) + 594*(5*(x*e + d)^(7/2) - 21*(
x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a*c*d^2 + 693*(3*(x*e + d)^(5/2) - 10*(x*e +
d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^2*d*e^2 + 22*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)
^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a*c*d + 99*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/
2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^2*e^2)*e^(-1)

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maple [A]  time = 0.05, size = 73, normalized size = 0.88 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (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}\right )}{693 e^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^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 = 1.06, size = 80, normalized size = 0.96 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.64, size = 80, normalized size = 0.96 \begin {gather*} \frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (99\,a^2\,e^4+99\,c^2\,d^4+63\,c^2\,d^2\,{\left (d+e\,x\right )}^2-154\,c^2\,d^3\,\left (d+e\,x\right )-198\,a\,c\,d^2\,e^2+154\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{693\,e^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(99*a^2*e^4 + 99*c^2*d^4 + 63*c^2*d^2*(d + e*x)^2 - 154*c^2*d^3*(d + e*x) - 198*a*c*d^2*e^2
+ 154*a*c*d*e^2*(d + e*x)))/(693*e^3)

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sympy [A]  time = 4.40, size = 97, normalized size = 1.17 \begin {gather*} \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 {gather*}

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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