\(\int \frac {(d+e x) (a+c x^2)}{(f+g x)^{3/2}} \, dx\) [598]

Optimal result

Integrand size = 22, antiderivative size = 111 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 (e f-d g) \left (c f^2+a g^2\right )}{g^4 \sqrt {f+g x}}+\frac {2 \left (a e g^2+c f (3 e f-2 d g)\right ) \sqrt {f+g x}}{g^4}-\frac {2 c (3 e f-d g) (f+g x)^{3/2}}{3 g^4}+\frac {2 c e (f+g x)^{5/2}}{5 g^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 111, 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} \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \left (a g^2+c f^2\right ) (e f-d g)}{g^4 \sqrt {f+g x}}+\frac {2 \sqrt {f+g x} \left (a e g^2+c f (3 e f-2 d g)\right )}{g^4}-\frac {2 c (f+g x)^{3/2} (3 e f-d g)}{3 g^4}+\frac {2 c e (f+g x)^{5/2}}{5 g^4} \]

[In]

Int[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {30 a g^2 (2 e f-d g+e g x)+10 c d g \left (-8 f^2-4 f g x+g^2 x^2\right )+6 c e \left (16 f^3+8 f^2 g x-2 f g^2 x^2+g^3 x^3\right )}{15 g^4 \sqrt {f+g x}} \]

[In]

Integrate[((d + e*x)*(a + c*x^2))/(f + g*x)^(3/2),x]

[Out]

(30*a*g^2*(2*e*f - d*g + e*g*x) + 10*c*d*g*(-8*f^2 - 4*f*g*x + g^2*x^2) + 6*c*e*(16*f^3 + 8*f^2*g*x - 2*f*g^2*
x^2 + g^3*x^3))/(15*g^4*Sqrt[f + g*x])

Maple [A] (verified)

Time = 0.43 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.77

method	result	size
pseudoelliptic	\(\frac {\left (\left (6 e \,x^{3}+10 d \,x^{2}\right ) c -30 a \left (-e x +d \right )\right ) g^{3}+60 \left (\left (-\frac {1}{5} e \,x^{2}-\frac {2}{3} d x \right ) c +a e \right ) f \,g^{2}-80 c \,f^{2} \left (-\frac {3 e x}{5}+d \right ) g +96 c e \,f^{3}}{15 \sqrt {g x +f}\, g^{4}}\)	\(86\)
gosper	\(-\frac {2 \left (-3 c e \,x^{3} g^{3}-5 c d \,g^{3} x^{2}+6 c e f \,g^{2} x^{2}-15 a e \,g^{3} x +20 c d f \,g^{2} x -24 c e \,f^{2} g x +15 a d \,g^{3}-30 a e f \,g^{2}+40 c d \,f^{2} g -48 c e \,f^{3}\right )}{15 \sqrt {g x +f}\, g^{4}}\)	\(101\)
trager	\(-\frac {2 \left (-3 c e \,x^{3} g^{3}-5 c d \,g^{3} x^{2}+6 c e f \,g^{2} x^{2}-15 a e \,g^{3} x +20 c d f \,g^{2} x -24 c e \,f^{2} g x +15 a d \,g^{3}-30 a e f \,g^{2}+40 c d \,f^{2} g -48 c e \,f^{3}\right )}{15 \sqrt {g x +f}\, g^{4}}\)	\(101\)
risch	\(\frac {2 \left (3 c e \,x^{2} g^{2}+5 c d x \,g^{2}-9 c e f g x +15 a e \,g^{2}-25 c d f g +33 c e \,f^{2}\right ) \sqrt {g x +f}}{15 g^{4}}-\frac {2 \left (a d \,g^{3}-a e f \,g^{2}+c d \,f^{2} g -c e \,f^{3}\right )}{g^{4} \sqrt {g x +f}}\)	\(101\)
derivativedivides	\(\frac {\frac {2 c e \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 c d g \left (g x +f \right )^{\frac {3}{2}}}{3}-2 c e f \left (g x +f \right )^{\frac {3}{2}}+2 a e \,g^{2} \sqrt {g x +f}-4 c d f g \sqrt {g x +f}+6 c e \,f^{2} \sqrt {g x +f}-\frac {2 \left (a d \,g^{3}-a e f \,g^{2}+c d \,f^{2} g -c e \,f^{3}\right )}{\sqrt {g x +f}}}{g^{4}}\)	\(120\)
default	\(\frac {\frac {2 c e \left (g x +f \right )^{\frac {5}{2}}}{5}+\frac {2 c d g \left (g x +f \right )^{\frac {3}{2}}}{3}-2 c e f \left (g x +f \right )^{\frac {3}{2}}+2 a e \,g^{2} \sqrt {g x +f}-4 c d f g \sqrt {g x +f}+6 c e \,f^{2} \sqrt {g x +f}-\frac {2 \left (a d \,g^{3}-a e f \,g^{2}+c d \,f^{2} g -c e \,f^{3}\right )}{\sqrt {g x +f}}}{g^{4}}\)	\(120\)

[In]

int((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/15*(((6*e*x^3+10*d*x^2)*c-30*a*(-e*x+d))*g^3+60*((-1/5*e*x^2-2/3*d*x)*c+a*e)*f*g^2-80*c*f^2*(-3/5*e*x+d)*g+9
6*c*e*f^3)/(g*x+f)^(1/2)/g^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (3 \, c e g^{3} x^{3} + 48 \, c e f^{3} - 40 \, c d f^{2} g + 30 \, a e f g^{2} - 15 \, a d g^{3} - {\left (6 \, c e f g^{2} - 5 \, c d g^{3}\right )} x^{2} + {\left (24 \, c e f^{2} g - 20 \, c d f g^{2} + 15 \, a e g^{3}\right )} x\right )} \sqrt {g x + f}}{15 \, {\left (g^{5} x + f g^{4}\right )}} \]

[In]

integrate((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

2/15*(3*c*e*g^3*x^3 + 48*c*e*f^3 - 40*c*d*f^2*g + 30*a*e*f*g^2 - 15*a*d*g^3 - (6*c*e*f*g^2 - 5*c*d*g^3)*x^2 +
(24*c*e*f^2*g - 20*c*d*f*g^2 + 15*a*e*g^3)*x)*sqrt(g*x + f)/(g^5*x + f*g^4)

Sympy [A] (verification not implemented)

Time = 1.97 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.35 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c e \left (f + g x\right )^{\frac {5}{2}}}{5 g^{3}} + \frac {\left (f + g x\right )^{\frac {3}{2}} \left (c d g - 3 c e f\right )}{3 g^{3}} + \frac {\sqrt {f + g x} \left (a e g^{2} - 2 c d f g + 3 c e f^{2}\right )}{g^{3}} - \frac {\left (a g^{2} + c f^{2}\right ) \left (d g - e f\right )}{g^{3} \sqrt {f + g x}}\right )}{g} & \text {for}\: g \neq 0 \\\frac {a d x + \frac {a e x^{2}}{2} + \frac {c d x^{3}}{3} + \frac {c e x^{4}}{4}}{f^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[In]

integrate((e*x+d)*(c*x**2+a)/(g*x+f)**(3/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (g x + f\right )}^{\frac {5}{2}} c e - 5 \, {\left (3 \, c e f - c d g\right )} {\left (g x + f\right )}^{\frac {3}{2}} + 15 \, {\left (3 \, c e f^{2} - 2 \, c d f g + a e g^{2}\right )} \sqrt {g x + f}}{g^{3}} + \frac {15 \, {\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )}}{\sqrt {g x + f} g^{3}}\right )}}{15 \, g} \]

[In]

integrate((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.23 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \, {\left (c e f^{3} - c d f^{2} g + a e f g^{2} - a d g^{3}\right )}}{\sqrt {g x + f} g^{4}} + \frac {2 \, {\left (3 \, {\left (g x + f\right )}^{\frac {5}{2}} c e g^{16} - 15 \, {\left (g x + f\right )}^{\frac {3}{2}} c e f g^{16} + 45 \, \sqrt {g x + f} c e f^{2} g^{16} + 5 \, {\left (g x + f\right )}^{\frac {3}{2}} c d g^{17} - 30 \, \sqrt {g x + f} c d f g^{17} + 15 \, \sqrt {g x + f} a e g^{18}\right )}}{15 \, g^{20}} \]

[In]

integrate((e*x+d)*(c*x^2+a)/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

2*(c*e*f^3 - c*d*f^2*g + a*e*f*g^2 - a*d*g^3)/(sqrt(g*x + f)*g^4) + 2/15*(3*(g*x + f)^(5/2)*c*e*g^16 - 15*(g*x
+ f)^(3/2)*c*e*f*g^16 + 45*sqrt(g*x + f)*c*e*f^2*g^16 + 5*(g*x + f)^(3/2)*c*d*g^17 - 30*sqrt(g*x + f)*c*d*f*g
^17 + 15*sqrt(g*x + f)*a*e*g^18)/g^20

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x) \left (a+c x^2\right )}{(f+g x)^{3/2}} \, dx=\frac {\sqrt {f+g\,x}\,\left (6\,c\,e\,f^2-4\,c\,d\,f\,g+2\,a\,e\,g^2\right )}{g^4}-\frac {-2\,c\,e\,f^3+2\,c\,d\,f^2\,g-2\,a\,e\,f\,g^2+2\,a\,d\,g^3}{g^4\,\sqrt {f+g\,x}}+\frac {2\,c\,e\,{\left (f+g\,x\right )}^{5/2}}{5\,g^4}+\frac {2\,c\,{\left (f+g\,x\right )}^{3/2}\,\left (d\,g-3\,e\,f\right )}{3\,g^4} \]

[In]

int(((a + c*x^2)*(d + e*x))/(f + g*x)^(3/2),x)

[Out]

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