\(\int \frac {a+c x^2}{\sqrt {d+e x}} \, dx\) [594]

Optimal result

Integrand size = 17, antiderivative size = 61 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {4 c d (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \]

[Out]

-4/3*c*d*(e*x+d)^(3/2)/e^3+2/5*c*(e*x+d)^(5/2)/e^3+2*(a*e^2+c*d^2)*(e*x+d)^(1/2)/e^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 61, 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} \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3}-\frac {4 c d (d+e x)^{3/2}}{3 e^3} \]

[In]

Int[(a + c*x^2)/Sqrt[d + e*x],x]

[Out]

(2*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^3 - (4*c*d*(d + e*x)^(3/2))/(3*e^3) + (2*c*(d + e*x)^(5/2))/(5*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 {c d^2+a e^2}{e^2 \sqrt {d+e x}}-\frac {2 c d \sqrt {d+e x}}{e^2}+\frac {c (d+e x)^{3/2}}{e^2}\right ) \, dx \\ & = \frac {2 \left (c d^2+a e^2\right ) \sqrt {d+e x}}{e^3}-\frac {4 c d (d+e x)^{3/2}}{3 e^3}+\frac {2 c (d+e x)^{5/2}}{5 e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (15 a e^2+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15*a*e^2 + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(\frac {2 \left (3 \left (c \,x^{2}+5 a \right ) e^{2}-4 x c d e +8 c \,d^{2}\right ) \sqrt {e x +d}}{15 e^{3}}\)	\(40\)
gosper	\(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}-4 x c d e +15 e^{2} a +8 c \,d^{2}\right )}{15 e^{3}}\)	\(41\)
trager	\(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}-4 x c d e +15 e^{2} a +8 c \,d^{2}\right )}{15 e^{3}}\)	\(41\)
risch	\(\frac {2 \sqrt {e x +d}\, \left (3 c \,x^{2} e^{2}-4 x c d e +15 e^{2} a +8 c \,d^{2}\right )}{15 e^{3}}\)	\(41\)
derivativedivides	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\)	\(52\)
default	\(\frac {\frac {2 c \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {4 c d \left (e x +d \right )^{\frac {3}{2}}}{3}+2 a \,e^{2} \sqrt {e x +d}+2 c \,d^{2} \sqrt {e x +d}}{e^{3}}\)	\(52\)

[In]

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

[Out]

2/15*(3*(c*x^2+5*a)*e^2-4*x*c*d*e+8*c*d^2)*(e*x+d)^(1/2)/e^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (3 \, c e^{2} x^{2} - 4 \, c d e x + 8 \, c d^{2} + 15 \, a e^{2}\right )} \sqrt {e x + d}}{15 \, e^{3}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.15 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 a \sqrt {d + e x} + \frac {2 c \left (d^{2} \sqrt {d + e x} - \frac {2 d \left (d + e x\right )^{\frac {3}{2}}}{3} + \frac {\left (d + e x\right )^{\frac {5}{2}}}{5}\right )}{e^{2}}}{e} & \text {for}\: e \neq 0 \\\frac {a x + \frac {c x^{3}}{3}}{\sqrt {d}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.87 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (15 \, \sqrt {e x + d} a + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} c}{e^{2}}\right )}}{15 \, e} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.72 \[ \int \frac {a+c x^2}{\sqrt {d+e x}} \, dx=\frac {2\,\sqrt {d+e\,x}\,\left (3\,c\,{\left (d+e\,x\right )}^2+15\,a\,e^2+15\,c\,d^2-10\,c\,d\,\left (d+e\,x\right )\right )}{15\,e^3} \]

[In]

int((a + c*x^2)/(d + e*x)^(1/2),x)

[Out]

(2*(d + e*x)^(1/2)*(3*c*(d + e*x)^2 + 15*a*e^2 + 15*c*d^2 - 10*c*d*(d + e*x)))/(15*e^3)