\(\int \sqrt {d+e x} (a d e+(c d^2+a e^2) x+c d e x^2) \, dx\) [1977]

Optimal result

Integrand size = 35, antiderivative size = 43 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2}{5} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \]

[Out]

2/5*(a-c*d^2/e^2)*(e*x+d)^(5/2)+2/7*c*d*(e*x+d)^(7/2)/e^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45} \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2}{5} (d+e x)^{5/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{7/2}}{7 e^2} \]

[In]

Int[Sqrt[d + e*x]*(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)^(5/2))/5 + (2*c*d*(d + e*x)^(7/2))/(7*e^2)

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2 (d+e x)^{5/2} \left (7 a e^2+c d (-2 d+5 e x)\right )}{35 e^2} \]

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(7*a*e^2 + c*d*(-2*d + 5*e*x)))/(35*e^2)

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 x c d e +7 e^{2} a -2 c \,d^{2}\right )}{35 e^{2}}\)	\(32\)
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (5 x c d e +7 e^{2} a -2 c \,d^{2}\right )}{35 e^{2}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\)	\(39\)
default	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}}{e^{2}}\)	\(39\)
trager	\(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\)	\(75\)
risch	\(\frac {2 \left (5 d \,e^{3} c \,x^{3}+7 e^{4} a \,x^{2}+8 d^{2} e^{2} c \,x^{2}+14 a d \,e^{3} x +c \,d^{3} e x +7 a \,d^{2} e^{2}-2 c \,d^{4}\right ) \sqrt {e x +d}}{35 e^{2}}\)	\(75\)

[In]

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

[Out]

2/35*(e*x+d)^(5/2)*(5*c*d*e*x+7*a*e^2-2*c*d^2)/e^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (35) = 70\).

Time = 0.35 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.72 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2 \, {\left (5 \, c d e^{3} x^{3} - 2 \, c d^{4} + 7 \, a d^{2} e^{2} + {\left (8 \, c d^{2} e^{2} + 7 \, a e^{4}\right )} x^{2} + {\left (c d^{3} e + 14 \, a d e^{3}\right )} x\right )} \sqrt {e x + d}}{35 \, e^{2}} \]

[In]

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

[Out]

2/35*(5*c*d*e^3*x^3 - 2*c*d^4 + 7*a*d^2*e^2 + (8*c*d^2*e^2 + 7*a*e^4)*x^2 + (c*d^3*e + 14*a*d*e^3)*x)*sqrt(e*x
+ d)/e^2

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.23 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\begin {cases} \frac {2 \left (\frac {c d \left (d + e x\right )^{\frac {7}{2}}}{7 e} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} & \text {for}\: e \neq 0 \\\frac {c d^{\frac {5}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(c*d*(d + e*x)**(7/2)/(7*e) + (d + e*x)**(5/2)*(a*e**2 - c*d**2)/(5*e))/e, Ne(e, 0)), (c*d**(5/2)
*x**2/2, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.88 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} c d - 7 \, {\left (c d^{2} - a e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}}\right )}}{35 \, e^{2}} \]

[In]

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

[Out]

2/35*(5*(e*x + d)^(7/2)*c*d - 7*(c*d^2 - a*e^2)*(e*x + d)^(5/2))/e^2

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (35) = 70\).

Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 4.60 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2 \, {\left (105 \, \sqrt {e x + d} a d^{2} e + \frac {35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} c d^{3}}{e} + 70 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a d e + \frac {14 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d^{2}}{e} + 7 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a e + \frac {3 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c d}{e}\right )}}{105 \, e} \]

[In]

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

[Out]

2/105*(105*sqrt(e*x + d)*a*d^2*e + 35*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*c*d^3/e + 70*((e*x + d)^(3/2) - 3*
sqrt(e*x + d)*d)*a*d*e + 14*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d^2/e + 7*(3*(
e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*e + 3*(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*d/e)/e

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx=\frac {2\,{\left (d+e\,x\right )}^{5/2}\,\left (7\,a\,e^2-7\,c\,d^2+5\,c\,d\,\left (d+e\,x\right )\right )}{35\,e^2} \]

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(7*a*e^2 - 7*c*d^2 + 5*c*d*(d + e*x)))/(35*e^2)