\(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2)^2 \, dx\) [1984]

Optimal result

Integrand size = 37, antiderivative size = 83 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \left (c d^2-a e^2\right )^2 (d+e x)^{7/2}}{7 e^3}-\frac {4 c d \left (c d^2-a e^2\right ) (d+e x)^{9/2}}{9 e^3}+\frac {2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45} \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=-\frac {4 c d (d+e x)^{9/2} \left (c d^2-a e^2\right )}{9 e^3}+\frac {2 (d+e x)^{7/2} \left (c d^2-a e^2\right )^2}{7 e^3}+\frac {2 c^2 d^2 (d+e x)^{11/2}}{11 e^3} \]

[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 45

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^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*} \text {integral}& = \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] (verified)

Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.81 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (99 a^2 e^4-22 a c d e^2 (2 d-7 e x)+c^2 d^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

[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)

Maple [A] (verified)

Time = 3.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.78

method	result	size
pseudoelliptic	\(\frac {2 \left (a^{2} e^{4}+\frac {14 x a c d \,e^{3}}{9}-\frac {4 \left (-\frac {63 c \,x^{2}}{44}+a \right ) c \,d^{2} e^{2}}{9}-\frac {28 x \,c^{2} d^{3} e}{99}+\frac {8 c^{2} d^{4}}{99}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{3}}\)	\(65\)
derivativedivides	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(68\)
default	\(\frac {\frac {2 c^{2} d^{2} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (e^{2} a -c \,d^{2}\right ) c d \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (e^{2} a -c \,d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{3}}\)	\(68\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (63 x^{2} c^{2} d^{2} e^{2}+154 x a c d \,e^{3}-28 x \,c^{2} d^{3} e +99 a^{2} e^{4}-44 a c \,d^{2} e^{2}+8 c^{2} d^{4}\right )}{693 e^{3}}\)	\(73\)
trager	\(\frac {2 \left (63 d^{2} e^{5} c^{2} x^{5}+154 a c d \,e^{6} x^{4}+161 c^{2} d^{3} e^{4} x^{4}+99 a^{2} e^{7} x^{3}+418 a \,d^{2} e^{5} c \,x^{3}+113 c^{2} d^{4} e^{3} x^{3}+297 a^{2} d \,e^{6} x^{2}+330 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+297 a^{2} d^{2} e^{5} x +22 a c \,d^{4} e^{3} x -4 c^{2} d^{6} e x +99 a^{2} d^{3} e^{4}-44 a c \,d^{5} e^{2}+8 c^{2} d^{7}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(192\)
risch	\(\frac {2 \left (63 d^{2} e^{5} c^{2} x^{5}+154 a c d \,e^{6} x^{4}+161 c^{2} d^{3} e^{4} x^{4}+99 a^{2} e^{7} x^{3}+418 a \,d^{2} e^{5} c \,x^{3}+113 c^{2} d^{4} e^{3} x^{3}+297 a^{2} d \,e^{6} x^{2}+330 a c \,d^{3} e^{4} x^{2}+3 c^{2} d^{5} e^{2} x^{2}+297 a^{2} d^{2} e^{5} x +22 a c \,d^{4} e^{3} x -4 c^{2} d^{6} e x +99 a^{2} d^{3} e^{4}-44 a c \,d^{5} e^{2}+8 c^{2} d^{7}\right ) \sqrt {e x +d}}{693 e^{3}}\)	\(192\)

[In]

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

[Out]

2/7*(a^2*e^4+14/9*x*a*c*d*e^3-4/9*(-63/44*c*x^2+a)*c*d^2*e^2-28/99*x*c^2*d^3*e+8/99*c^2*d^4)*(e*x+d)^(7/2)/e^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (71) = 142\).

Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.22 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (63 \, c^{2} d^{2} e^{5} x^{5} + 8 \, c^{2} d^{7} - 44 \, a c d^{5} e^{2} + 99 \, a^{2} d^{3} e^{4} + 7 \, {\left (23 \, c^{2} d^{3} e^{4} + 22 \, a c d e^{6}\right )} x^{4} + {\left (113 \, c^{2} d^{4} e^{3} + 418 \, a c d^{2} e^{5} + 99 \, a^{2} e^{7}\right )} x^{3} + 3 \, {\left (c^{2} d^{5} e^{2} + 110 \, a c d^{3} e^{4} + 99 \, a^{2} d e^{6}\right )} x^{2} - {\left (4 \, c^{2} d^{6} e - 22 \, a c d^{4} e^{3} - 297 \, a^{2} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{693 \, e^{3}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.33 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} d^{2} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{2}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (2 a c d e^{2} - 2 c^{2} d^{3}\right )}{9 e^{2}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{7 e^{2}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c^{2} d^{\frac {9}{2}} x^{3}}{3} & \text {otherwise} \end {cases} \]

[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]

Piecewise((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, Ne(e, 0)), (c**2*d**(9/2)*x**3/3, Tru
e))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} c^{2} d^{2} - 154 \, {\left (c^{2} d^{3} - a c d e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 99 \, {\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{693 \, e^{3}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (71) = 142\).

Time = 0.28 (sec) , antiderivative size = 566, normalized size of antiderivative = 6.82 \[ \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=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{2} d^{3} e^{2} + 2310 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a c d^{4} + 3465 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{2} d^{2} e^{2} + 1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a c d^{3} + \frac {231 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c^{2} d^{5}}{e^{2}} + 693 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} d e^{2} + 594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a c d^{2} + \frac {297 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c^{2} d^{4}}{e^{2}} + 99 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} e^{2} + 22 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a c d + \frac {33 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} c^{2} d^{3}}{e^{2}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} c^{2} d^{2}}{e^{2}}\right )}}{3465 \, e} \]

[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*(3465*sqrt(e*x + d)*a^2*d^3*e^2 + 2310*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a*c*d^4 + 3465*((e*x + d)^
(3/2) - 3*sqrt(e*x + d)*d)*a^2*d^2*e^2 + 1386*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2
)*a*c*d^3 + 231*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c^2*d^5/e^2 + 693*(3*(e*x +
d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*d*e^2 + 594*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2
)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a*c*d^2 + 297*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d +
35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*c^2*d^4/e^2 + 99*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 3
5*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^2*e^2 + 22*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(
e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*c*d + 33*(35*(e*x + d)^(9/2) - 180*(e*
x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*c^2*d^3/e^2 + 5*(6
3*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x +
d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*c^2*d^2/e^2)/e

Mupad [B] (verification not implemented)

Time = 9.87 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2 \, dx=\frac {2\,{\left (d+e\,x\right )}^{7/2}\,\left (99\,a^2\,e^4+99\,c^2\,d^4+63\,c^2\,d^2\,{\left (d+e\,x\right )}^2-154\,c^2\,d^3\,\left (d+e\,x\right )-198\,a\,c\,d^2\,e^2+154\,a\,c\,d\,e^2\,\left (d+e\,x\right )\right )}{693\,e^3} \]

[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)