\(\int (d+e x) \sqrt {b x+c x^2} \, dx\) [287]

Optimal result

Integrand size = 19, antiderivative size = 99 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {b^2 (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}} \]

[Out]

1/3*e*(c*x^2+b*x)^(3/2)/c-1/8*b^2*(-b*e+2*c*d)*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(5/2)+1/8*(-b*e+2*c*d)*(
2*c*x+b)*(c*x^2+b*x)^(1/2)/c^2

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {654, 626, 634, 212} \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=-\frac {b^2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) (2 c d-b e)}{8 c^{5/2}}+\frac {(b+2 c x) \sqrt {b x+c x^2} (2 c d-b e)}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c} \]

[In]

Int[(d + e*x)*Sqrt[b*x + c*x^2],x]

[Out]

((2*c*d - b*e)*(b + 2*c*x)*Sqrt[b*x + c*x^2])/(8*c^2) + (e*(b*x + c*x^2)^(3/2))/(3*c) - (b^2*(2*c*d - b*e)*Arc
Tanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(8*c^(5/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 626

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x)*((a + b*x + c*x^2)^p/(2*c*(2*p + 1
))), x] - Dist[p*((b^2 - 4*a*c)/(2*c*(2*p + 1))), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 634

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2
]], x] /; FreeQ[{b, c}, x]

Rule 654

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*((a + b*x + c*x^2)^(p +
1)/(2*c*(p + 1))), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {e \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {(2 c d-b e) \int \sqrt {b x+c x^2} \, dx}{2 c} \\ & = \frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {\left (b^2 (2 c d-b e)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^2} \\ & = \frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {\left (b^2 (2 c d-b e)\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^2} \\ & = \frac {(2 c d-b e) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {e \left (b x+c x^2\right )^{3/2}}{3 c}-\frac {b^2 (2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\frac {\sqrt {x (b+c x)} \left (\sqrt {c} \sqrt {x} \left (-3 b^2 e+2 b c (3 d+e x)+4 c^2 x (3 d+2 e x)\right )+\frac {6 b^2 (-2 c d+b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )}{\sqrt {b+c x}}\right )}{24 c^{5/2} \sqrt {x}} \]

[In]

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

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*Sqrt[x]*(-3*b^2*e + 2*b*c*(3*d + e*x) + 4*c^2*x*(3*d + 2*e*x)) + (6*b^2*(-2*c*d +
b*e)*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])])/Sqrt[b + c*x]))/(24*c^(5/2)*Sqrt[x])

Maple [A] (verified)

Time = 1.89 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\left (b^{3} e -2 b^{2} c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\left (-2 \left (\frac {e x}{3}+d \right ) b \,c^{\frac {3}{2}}+\left (-\frac {8}{3} e \,x^{2}-4 d x \right ) c^{\frac {5}{2}}+\sqrt {c}\, b^{2} e \right ) \sqrt {x \left (c x +b \right )}}{8 c^{\frac {5}{2}}}\)	\(85\)
risch	\(-\frac {\left (-8 c^{2} e \,x^{2}-2 b c e x -12 c^{2} d x +3 b^{2} e -6 b c d \right ) x \left (c x +b \right )}{24 c^{2} \sqrt {x \left (c x +b \right )}}+\frac {b^{2} \left (b e -2 c d \right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {5}{2}}}\)	\(96\)
default	\(d \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )+e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\)	\(139\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.07 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\left [-\frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \, {\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{3}}, \frac {3 \, {\left (2 \, b^{2} c d - b^{3} e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} e x^{2} + 6 \, b c^{2} d - 3 \, b^{2} c e + 2 \, {\left (6 \, c^{3} d + b c^{2} e\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{3}}\right ] \]

[In]

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

[Out]

[-1/48*(3*(2*b^2*c*d - b^3*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 2*(8*c^3*e*x^2 + 6*b*c^2*
d - 3*b^2*c*e + 2*(6*c^3*d + b*c^2*e)*x)*sqrt(c*x^2 + b*x))/c^3, 1/24*(3*(2*b^2*c*d - b^3*e)*sqrt(-c)*arctan(s
qrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (8*c^3*e*x^2 + 6*b*c^2*d - 3*b^2*c*e + 2*(6*c^3*d + b*c^2*e)*x)*sqrt(c*x^2
+ b*x))/c^3]

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.69 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\begin {cases} - \frac {b \left (b d - \frac {3 b \left (\frac {b e}{6} + c d\right )}{4 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{2 c} + \sqrt {b x + c x^{2}} \left (\frac {e x^{2}}{3} + \frac {x \left (\frac {b e}{6} + c d\right )}{2 c} + \frac {b d - \frac {3 b \left (\frac {b e}{6} + c d\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (\frac {d \left (b x\right )^{\frac {3}{2}}}{3} + \frac {e \left (b x\right )^{\frac {5}{2}}}{5 b}\right )}{b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-b*(b*d - 3*b*(b*e/6 + c*d)/(4*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c)
, Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True))/(2*c) + sqrt(b*x + c*x**2)*
(e*x**2/3 + x*(b*e/6 + c*d)/(2*c) + (b*d - 3*b*(b*e/6 + c*d)/(4*c))/c), Ne(c, 0)), (2*(d*(b*x)**(3/2)/3 + e*(b
*x)**(5/2)/(5*b))/b, Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.56 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\frac {1}{2} \, \sqrt {c x^{2} + b x} d x - \frac {\sqrt {c x^{2} + b x} b e x}{4 \, c} - \frac {b^{2} d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} + \frac {b^{3} e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} + \frac {\sqrt {c x^{2} + b x} b d}{4 \, c} - \frac {\sqrt {c x^{2} + b x} b^{2} e}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} e}{3 \, c} \]

[In]

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

[Out]

1/2*sqrt(c*x^2 + b*x)*d*x - 1/4*sqrt(c*x^2 + b*x)*b*e*x/c - 1/8*b^2*d*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt
(c))/c^(3/2) + 1/16*b^3*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(5/2) + 1/4*sqrt(c*x^2 + b*x)*b*d/c -
1/8*sqrt(c*x^2 + b*x)*b^2*e/c^2 + 1/3*(c*x^2 + b*x)^(3/2)*e/c

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.03 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=\frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, e x + \frac {6 \, c^{2} d + b c e}{c^{2}}\right )} x + \frac {3 \, {\left (2 \, b c d - b^{2} e\right )}}{c^{2}}\right )} + \frac {{\left (2 \, b^{2} c d - b^{3} e\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {5}{2}}} \]

[In]

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

[Out]

1/24*sqrt(c*x^2 + b*x)*(2*(4*e*x + (6*c^2*d + b*c*e)/c^2)*x + 3*(2*b*c*d - b^2*e)/c^2) + 1/16*(2*b^2*c*d - b^3
*e)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(5/2)

Mupad [B] (verification not implemented)

Time = 9.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.28 \[ \int (d+e x) \sqrt {b x+c x^2} \, dx=d\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )-\frac {b^2\,d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}}+\frac {b^3\,e\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {e\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2} \]

[In]

int((b*x + c*x^2)^(1/2)*(d + e*x),x)

[Out]

d*(b*x + c*x^2)^(1/2)*(x/2 + b/(4*c)) - (b^2*d*log((b/2 + c*x)/c^(1/2) + (b*x + c*x^2)^(1/2)))/(8*c^(3/2)) + (
b^3*e*log((b + 2*c*x)/c^(1/2) + 2*(b*x + c*x^2)^(1/2)))/(16*c^(5/2)) + (e*(b*x + c*x^2)^(1/2)*(8*c^2*x^2 - 3*b
^2 + 2*b*c*x))/(24*c^2)