\(\int \frac {(d+e x) (a+c x^2)}{\sqrt {f+g x}} \, dx\) [591]

Optimal result

Integrand size = 22, antiderivative size = 113 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {2 (e f-d g) \left (c f^2+a g^2\right ) \sqrt {f+g x}}{g^4}+\frac {2 \left (a e g^2+c f (3 e f-2 d g)\right ) (f+g x)^{3/2}}{3 g^4}-\frac {2 c (3 e f-d g) (f+g x)^{5/2}}{5 g^4}+\frac {2 c e (f+g x)^{7/2}}{7 g^4} \]

[Out]

2/3*(a*e*g^2+c*f*(-2*d*g+3*e*f))*(g*x+f)^(3/2)/g^4-2/5*c*(-d*g+3*e*f)*(g*x+f)^(5/2)/g^4+2/7*c*e*(g*x+f)^(7/2)/
g^4-2*(-d*g+e*f)*(a*g^2+c*f^2)*(g*x+f)^(1/2)/g^4

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {786} \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=-\frac {2 \sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g)}{g^4}+\frac {2 (f+g x)^{3/2} \left (a e g^2+c f (3 e f-2 d g)\right )}{3 g^4}-\frac {2 c (f+g x)^{5/2} (3 e f-d g)}{5 g^4}+\frac {2 c e (f+g x)^{7/2}}{7 g^4} \]

[In]

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

[Out]

(-2*(e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x])/g^4 + (2*(a*e*g^2 + c*f*(3*e*f - 2*d*g))*(f + g*x)^(3/2))/(3*g^
4) - (2*c*(3*e*f - d*g)*(f + g*x)^(5/2))/(5*g^4) + (2*c*e*(f + g*x)^(7/2))/(7*g^4)

Rule 786

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

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (35 a g^2 (-2 e f+3 d g+e g x)+7 c d g \left (8 f^2-4 f g x+3 g^2 x^2\right )-3 c e \left (16 f^3-8 f^2 g x+6 f g^2 x^2-5 g^3 x^3\right )\right )}{105 g^4} \]

[In]

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

[Out]

(2*Sqrt[f + g*x]*(35*a*g^2*(-2*e*f + 3*d*g + e*g*x) + 7*c*d*g*(8*f^2 - 4*f*g*x + 3*g^2*x^2) - 3*c*e*(16*f^3 -
8*f^2*g*x + 6*f*g^2*x^2 - 5*g^3*x^3)))/(105*g^4)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {g x +f}\, \left (\left (\frac {x^{2} \left (\frac {5 e x}{7}+d \right ) c}{5}+a \left (\frac {e x}{3}+d \right )\right ) g^{3}-\frac {2 f \left (\frac {2 \left (\frac {9 e x}{14}+d \right ) x c}{5}+a e \right ) g^{2}}{3}+\frac {8 c \left (\frac {3 e x}{7}+d \right ) f^{2} g}{15}-\frac {16 c e \,f^{3}}{35}\right )}{g^{4}}\)	\(79\)
gosper	\(\frac {2 \sqrt {g x +f}\, \left (15 c e \,x^{3} g^{3}+21 c d \,g^{3} x^{2}-18 c e f \,g^{2} x^{2}+35 a e \,g^{3} x -28 c d f \,g^{2} x +24 c e \,f^{2} g x +105 a d \,g^{3}-70 a e f \,g^{2}+56 c d \,f^{2} g -48 c e \,f^{3}\right )}{105 g^{4}}\)	\(101\)
trager	\(\frac {2 \sqrt {g x +f}\, \left (15 c e \,x^{3} g^{3}+21 c d \,g^{3} x^{2}-18 c e f \,g^{2} x^{2}+35 a e \,g^{3} x -28 c d f \,g^{2} x +24 c e \,f^{2} g x +105 a d \,g^{3}-70 a e f \,g^{2}+56 c d \,f^{2} g -48 c e \,f^{3}\right )}{105 g^{4}}\)	\(101\)
risch	\(\frac {2 \sqrt {g x +f}\, \left (15 c e \,x^{3} g^{3}+21 c d \,g^{3} x^{2}-18 c e f \,g^{2} x^{2}+35 a e \,g^{3} x -28 c d f \,g^{2} x +24 c e \,f^{2} g x +105 a d \,g^{3}-70 a e f \,g^{2}+56 c d \,f^{2} g -48 c e \,f^{3}\right )}{105 g^{4}}\)	\(101\)
derivativedivides	\(\frac {\frac {2 c e \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right ) c -2 c e f \right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-2 \left (d g -e f \right ) c f +e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) \sqrt {g x +f}}{g^{4}}\)	\(105\)
default	\(\frac {\frac {2 c e \left (g x +f \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (d g -e f \right ) c -2 c e f \right ) \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 \left (-2 \left (d g -e f \right ) c f +e \left (a \,g^{2}+c \,f^{2}\right )\right ) \left (g x +f \right )^{\frac {3}{2}}}{3}+2 \left (d g -e f \right ) \left (a \,g^{2}+c \,f^{2}\right ) \sqrt {g x +f}}{g^{4}}\)	\(105\)

[In]

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

[Out]

2*(g*x+f)^(1/2)*((1/5*x^2*(5/7*e*x+d)*c+a*(1/3*e*x+d))*g^3-2/3*f*(2/5*(9/14*e*x+d)*x*c+a*e)*g^2+8/15*c*(3/7*e*
x+d)*f^2*g-16/35*c*e*f^3)/g^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (15 \, c e g^{3} x^{3} - 48 \, c e f^{3} + 56 \, c d f^{2} g - 70 \, a e f g^{2} + 105 \, a d g^{3} - 3 \, {\left (6 \, c e f g^{2} - 7 \, c d g^{3}\right )} x^{2} + {\left (24 \, c e f^{2} g - 28 \, c d f g^{2} + 35 \, a e g^{3}\right )} x\right )} \sqrt {g x + f}}{105 \, g^{4}} \]

[In]

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

[Out]

2/105*(15*c*e*g^3*x^3 - 48*c*e*f^3 + 56*c*d*f^2*g - 70*a*e*f*g^2 + 105*a*d*g^3 - 3*(6*c*e*f*g^2 - 7*c*d*g^3)*x
^2 + (24*c*e*f^2*g - 28*c*d*f*g^2 + 35*a*e*g^3)*x)*sqrt(g*x + f)/g^4

Sympy [A] (verification not implemented)

Time = 0.72 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\begin {cases} \frac {2 \left (\frac {c e \left (f + g x\right )^{\frac {7}{2}}}{7 g^{3}} + \frac {\left (f + g x\right )^{\frac {5}{2}} \left (c d g - 3 c e f\right )}{5 g^{3}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (a e g^{2} - 2 c d f g + 3 c e f^{2}\right )}{3 g^{3}} + \frac {\sqrt {f + g x} \left (a d g^{3} - a e f g^{2} + c d f^{2} g - c e f^{3}\right )}{g^{3}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d x + \frac {a e x^{2}}{2} + \frac {c d x^{3}}{3} + \frac {c e x^{4}}{4}}{\sqrt {f}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*(c*e*(f + g*x)**(7/2)/(7*g**3) + (f + g*x)**(5/2)*(c*d*g - 3*c*e*f)/(5*g**3) + (f + g*x)**(3/2)*(
a*e*g**2 - 2*c*d*f*g + 3*c*e*f**2)/(3*g**3) + sqrt(f + g*x)*(a*d*g**3 - a*e*f*g**2 + c*d*f**2*g - c*e*f**3)/g*
*3)/g, Ne(g, 0)), ((a*d*x + a*e*x**2/2 + c*d*x**3/3 + c*e*x**4/4)/sqrt(f), True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.92 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (15 \, {\left (g x + f\right )}^{\frac {7}{2}} c e - 21 \, {\left (3 \, c e f - c d g\right )} {\left (g x + f\right )}^{\frac {5}{2}} + 35 \, {\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )} {\left (g x + f\right )}^{\frac {3}{2}} - 105 \, {\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )} \sqrt {g x + f}\right )}}{105 \, g^{4}} \]

[In]

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

[Out]

2/105*(15*(g*x + f)^(7/2)*c*e - 21*(3*c*e*f - c*d*g)*(g*x + f)^(5/2) + 35*(3*c*e*f^2 - 2*c*d*f*g + a*e*g^2)*(g
*x + f)^(3/2) - 105*(c*e*f^3 - c*d*f^2*g + a*e*f*g^2 - a*d*g^3)*sqrt(g*x + f))/g^4

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (105 \, \sqrt {g x + f} a d + \frac {35 \, {\left ({\left (g x + f\right )}^{\frac {3}{2}} - 3 \, \sqrt {g x + f} f\right )} a e}{g} + \frac {7 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} - 10 \, {\left (g x + f\right )}^{\frac {3}{2}} f + 15 \, \sqrt {g x + f} f^{2}\right )} c d}{g^{2}} + \frac {3 \, {\left (5 \, {\left (g x + f\right )}^{\frac {7}{2}} - 21 \, {\left (g x + f\right )}^{\frac {5}{2}} f + 35 \, {\left (g x + f\right )}^{\frac {3}{2}} f^{2} - 35 \, \sqrt {g x + f} f^{3}\right )} c e}{g^{3}}\right )}}{105 \, g} \]

[In]

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

[Out]

2/105*(105*sqrt(g*x + f)*a*d + 35*((g*x + f)^(3/2) - 3*sqrt(g*x + f)*f)*a*e/g + 7*(3*(g*x + f)^(5/2) - 10*(g*x
+ f)^(3/2)*f + 15*sqrt(g*x + f)*f^2)*c*d/g^2 + 3*(5*(g*x + f)^(7/2) - 21*(g*x + f)^(5/2)*f + 35*(g*x + f)^(3/
2)*f^2 - 35*sqrt(g*x + f)*f^3)*c*e/g^3)/g

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{\sqrt {f+g x}} \, dx=\frac {{\left (f+g\,x\right )}^{3/2}\,\left (6\,c\,e\,f^2-4\,c\,d\,f\,g+2\,a\,e\,g^2\right )}{3\,g^4}+\frac {2\,c\,e\,{\left (f+g\,x\right )}^{7/2}}{7\,g^4}+\frac {2\,c\,{\left (f+g\,x\right )}^{5/2}\,\left (d\,g-3\,e\,f\right )}{5\,g^4}+\frac {2\,\sqrt {f+g\,x}\,\left (c\,f^2+a\,g^2\right )\,\left (d\,g-e\,f\right )}{g^4} \]

[In]

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

[Out]

((f + g*x)^(3/2)*(2*a*e*g^2 + 6*c*e*f^2 - 4*c*d*f*g))/(3*g^4) + (2*c*e*(f + g*x)^(7/2))/(7*g^4) + (2*c*(f + g*
x)^(5/2)*(d*g - 3*e*f))/(5*g^4) + (2*(f + g*x)^(1/2)*(a*g^2 + c*f^2)*(d*g - e*f))/g^4