\(\int \sqrt {d+e x} (a+c x^2) \, dx\) [593]

Optimal result

Integrand size = 17, antiderivative size = 63 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}{3 e^3}-\frac {4 c d (d+e x)^{5/2}}{5 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^3}+\frac {2 c (d+e x)^{7/2}}{7 e^3}-\frac {4 c d (d+e x)^{5/2}}{5 e^3} \]

[In]

Int[Sqrt[d + e*x]*(a + c*x^2),x]

[Out]

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

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 (d+e x)^{3/2} \left (35 a e^2+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(35*a*e^2 + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)

Maple [A] (verified)

Time = 2.05 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.63

method	result	size
pseudoelliptic	\(\frac {\left (\left (30 c \,x^{2}+70 a \right ) e^{2}-24 x c d e +16 c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{105 e^{3}}\)	\(40\)
gosper	\(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (15 c \,x^{2} e^{2}-12 x c d e +35 e^{2} a +8 c \,d^{2}\right )}{105 e^{3}}\)	\(41\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {4 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(48\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {7}{2}}}{7}-\frac {4 c d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (e^{2} a +c \,d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{3}}\)	\(48\)
trager	\(\frac {2 \left (15 c \,x^{3} e^{3}+3 c d \,x^{2} e^{2}+35 a \,e^{3} x -4 c \,d^{2} e x +35 a d \,e^{2}+8 d^{3} c \right ) \sqrt {e x +d}}{105 e^{3}}\)	\(61\)
risch	\(\frac {2 \left (15 c \,x^{3} e^{3}+3 c d \,x^{2} e^{2}+35 a \,e^{3} x -4 c \,d^{2} e x +35 a d \,e^{2}+8 d^{3} c \right ) \sqrt {e x +d}}{105 e^{3}}\)	\(61\)

[In]

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

[Out]

1/105*((30*c*x^2+70*a)*e^2-24*x*c*d*e+16*c*d^2)*(e*x+d)^(3/2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.98 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (15 \, c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 8 \, c d^{3} + 35 \, a d e^{2} - {\left (4 \, c d^{2} e - 35 \, a e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, e^{3}} \]

[In]

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

[Out]

2/105*(15*c*e^3*x^3 + 3*c*d*e^2*x^2 + 8*c*d^3 + 35*a*d*e^2 - (4*c*d^2*e - 35*a*e^3)*x)*sqrt(e*x + d)/e^3

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.21 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\begin {cases} \frac {2 \left (- \frac {2 c d \left (d + e x\right )^{\frac {5}{2}}}{5 e^{2}} + \frac {c \left (d + e x\right )^{\frac {7}{2}}}{7 e^{2}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a e^{2} + c d^{2}\right )}{3 e^{2}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a x + \frac {c x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(-2*c*d*(d + e*x)**(5/2)/(5*e**2) + c*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(3/2)*(a*e**2 + c*d*
*2)/(3*e**2))/e, Ne(e, 0)), (sqrt(d)*(a*x + c*x**3/3), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.75 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c - 42 \, {\left (e x + d\right )}^{\frac {5}{2}} c d + 35 \, {\left (c d^{2} + a e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{105 \, e^{3}} \]

[In]

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

[Out]

2/105*(15*(e*x + d)^(7/2)*c - 42*(e*x + d)^(5/2)*c*d + 35*(c*d^2 + a*e^2)*(e*x + d)^(3/2))/e^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (51) = 102\).

Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.02 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2 \, {\left (105 \, \sqrt {e x + d} a d + 35 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a + \frac {7 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c d}{e^{2}} + \frac {3 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} c}{e^{2}}\right )}}{105 \, e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.70 \[ \int \sqrt {d+e x} \left (a+c x^2\right ) \, dx=\frac {2\,{\left (d+e\,x\right )}^{3/2}\,\left (15\,c\,{\left (d+e\,x\right )}^2+35\,a\,e^2+35\,c\,d^2-42\,c\,d\,\left (d+e\,x\right )\right )}{105\,e^3} \]

[In]

int((a + c*x^2)*(d + e*x)^(1/2),x)

[Out]

(2*(d + e*x)^(3/2)*(15*c*(d + e*x)^2 + 35*a*e^2 + 35*c*d^2 - 42*c*d*(d + e*x)))/(105*e^3)