\(\int \frac {a d e+(c d^2+a e^2) x+c d e x^2}{\sqrt {d+e x}} \, dx\) [1978]

Optimal result

Integrand size = 35, antiderivative size = 43 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2}{3} \left (a-\frac {c d^2}{e^2}\right ) (d+e x)^{3/2}+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \]

[Out]

2/3*(a-c*d^2/e^2)*(e*x+d)^(3/2)+2/5*c*d*(e*x+d)^(5/2)/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} \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2}{3} (d+e x)^{3/2} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{5/2}}{5 e^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2 (d+e x)^{3/2} \left (5 a e^2+c d (-2 d+3 e x)\right )}{15 e^2} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(5*a*e^2 + c*d*(-2*d + 3*e*x)))/(15*e^2)

Maple [A] (verified)

Time = 2.74 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.74

method	result	size
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3 x c d e +5 e^{2} a -2 c \,d^{2}\right )}{15 e^{2}}\)	\(32\)
pseudoelliptic	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3 x c d e +5 e^{2} a -2 c \,d^{2}\right )}{15 e^{2}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\)	\(39\)
default	\(\frac {\frac {2 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a -c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{2}}\)	\(39\)
trager	\(\frac {2 \left (3 c d \,x^{2} e^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 d^{3} c \right ) \sqrt {e x +d}}{15 e^{2}}\)	\(51\)
risch	\(\frac {2 \left (3 c d \,x^{2} e^{2}+5 a \,e^{3} x +c \,d^{2} e x +5 a d \,e^{2}-2 d^{3} c \right ) \sqrt {e x +d}}{15 e^{2}}\)	\(51\)

[In]

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

[Out]

2/15*(e*x+d)^(3/2)*(3*c*d*e*x+5*a*e^2-2*c*d^2)/e^2

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.19 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, c d e^{2} x^{2} - 2 \, c d^{3} + 5 \, a d e^{2} + {\left (c d^{2} e + 5 \, a e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, e^{2}} \]

[In]

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

[Out]

2/15*(3*c*d*e^2*x^2 - 2*c*d^3 + 5*a*d*e^2 + (c*d^2*e + 5*a*e^3)*x)*sqrt(e*x + d)/e^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (42) = 84\).

Time = 0.65 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.42 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 a d e \sqrt {d + e x} + \frac {2 c d \left (d^{2} \sqrt {d + e x} - \frac {2 d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e} + \frac {2 \left (a e^{2} + c d^{2}\right ) \left (- d \sqrt {d + e x} + \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e}}{e} & \text {for}\: e \neq 0 \\\frac {c d^{\frac {3}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((2*a*d*e*sqrt(d + e*x) + 2*c*d*(d**2*sqrt(d + e*x) - 2*d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e
+ 2*(a*e**2 + c*d**2)*(-d*sqrt(d + e*x) + (d + e*x)**(3/2)/3)/e)/e, Ne(e, 0)), (c*d**(3/2)*x**2/2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (35) = 70\).

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.09 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a d e + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d}{e} + \frac {5 \, {\left (c d^{2} + a e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )}}{e}\right )}}{15 \, e} \]

[In]

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

[Out]

2/15*(15*sqrt(e*x + d)*a*d*e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e + 5*(c*
d^2 + a*e^2)*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)/e)/e

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (35) = 70\).

Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.44 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a d e + \frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} c d^{2}}{e} + 5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a e + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d}{e}\right )}}{15 \, e} \]

[In]

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

[Out]

2/15*(15*sqrt(e*x + d)*a*d*e + 5*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*c*d^2/e + 5*((e*x + d)^(3/2) - 3*sqrt(e
*x + d)*d)*a*e + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c*d/e)/e

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{\sqrt {d+e x}} \, dx=\frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (5\,a\,e^2-5\,c\,d^2+3\,c\,d\,\left (d+e\,x\right )\right )}{15\,e^2} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(5*a*e^2 - 5*c*d^2 + 3*c*d*(d + e*x)))/(15*e^2)