3.9.73 \(\int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx\) [873]

Optimal. Leaf size=131 \[ \frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}+\frac {(e f-d g) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e \sqrt {c d^2-b d e+a e^2}} \]

[Out]

g*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/e/c^(1/2)+(-d*g+e*f)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*
x)/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/e/(a*e^2-b*d*e+c*d^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {857, 635, 212, 738} \begin {gather*} \frac {(e f-d g) \tanh ^{-1}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e \sqrt {a e^2-b d e+c d^2}}+\frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) + ((e*f - d*g)*ArcTanh[(b*d - 2*a*e + (
2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(e*Sqrt[c*d^2 - b*d*e + a*e^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 635

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

Rule 738

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {f+g x}{(d+e x) \sqrt {a+b x+c x^2}} \, dx &=\frac {g \int \frac {1}{\sqrt {a+b x+c x^2}} \, dx}{e}+\frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {a+b x+c x^2}} \, dx}{e}\\ &=\frac {(2 g) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c x}{\sqrt {a+b x+c x^2}}\right )}{e}-\frac {(2 (e f-d g)) \text {Subst}\left (\int \frac {1}{4 c d^2-4 b d e+4 a e^2-x^2} \, dx,x,\frac {-b d+2 a e-(2 c d-b e) x}{\sqrt {a+b x+c x^2}}\right )}{e}\\ &=\frac {g \tanh ^{-1}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{\sqrt {c} e}+\frac {(e f-d g) \tanh ^{-1}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e \sqrt {c d^2-b d e+a e^2}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 142, normalized size = 1.08 \begin {gather*} -\frac {\frac {2 \sqrt {-c d^2+b d e-a e^2} (-e f+d g) \tan ^{-1}\left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )}{c d^2+e (-b d+a e)}+\frac {g \log \left (e \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{\sqrt {c}}}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(e*f) + d*g)*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[
-(c*d^2) + e*(b*d - a*e)]])/(c*d^2 + e*(-(b*d) + a*e)) + (g*Log[e*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]
)])/Sqrt[c])/e)

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Maple [A]
time = 0.12, size = 199, normalized size = 1.52

method	result	size
default	\(\frac {g \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}-\frac {\left (-d g +e f \right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (e b -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-(-d*g+e*f)/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(
a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d
/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((%e^-1*b-2*%e^-2*c*d)^2>0)', s
ee `assume?`

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (117) = 234\).
time = 196.39, size = 1073, normalized size = 8.19 \begin {gather*} \left [\frac {{\left (c d^{2} g - b d g e + a g e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) - \sqrt {c d^{2} - b d e + a e^{2}} {\left (c d g - c f e\right )} \log \left (-\frac {8 \, c^{2} d^{2} x^{2} + 8 \, b c d^{2} x + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c d^{2} - b d e + a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a} + {\left (8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{2} + 4 \, a b d + {\left (3 \, b^{2} + 4 \, a c\right )} d x\right )} e}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )}{2 \, {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}, -\frac {2 \, \sqrt {-c d^{2} + b d e - a e^{2}} {\left (c d g - c f e\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2} + {\left (a c x^{2} + a b x + a^{2}\right )} e^{2} - {\left (b c d x^{2} + b^{2} d x + a b d\right )} e\right )}}\right ) - {\left (c d^{2} g - b d g e + a g e^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right )}{2 \, {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}, -\frac {2 \, {\left (c d^{2} g - b d g e + a g e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + \sqrt {c d^{2} - b d e + a e^{2}} {\left (c d g - c f e\right )} \log \left (-\frac {8 \, c^{2} d^{2} x^{2} + 8 \, b c d^{2} x + {\left (b^{2} + 4 \, a c\right )} d^{2} + 4 \, \sqrt {c d^{2} - b d e + a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a} + {\left (8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} e^{2} - 2 \, {\left (4 \, b c d x^{2} + 4 \, a b d + {\left (3 \, b^{2} + 4 \, a c\right )} d x\right )} e}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )}{2 \, {\left (c^{2} d^{2} e - b c d e^{2} + a c e^{3}\right )}}, -\frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left (c d g - c f e\right )} \arctan \left (-\frac {\sqrt {-c d^{2} + b d e - a e^{2}} {\left (2 \, c d x + b d - {\left (b x + 2 \, a\right )} e\right )} \sqrt {c x^{2} + b x + a}}{2 \, {\left (c^{2} d^{2} x^{2} + b c d^{2} x + a c d^{2} + {\left (a c x^{2} + a b x + a^{2}\right )} e^{2} - {\left (b c d x^{2} + b^{2} d x + a b d\right )} e\right )}}\right ) + {\left (c d^{2} g - b d g e + a g e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{c^{2} d^{2} e - b c d e^{2} + a c e^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((c*d^2*g - b*d*g*e + a*g*e^2)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x +
b)*sqrt(c) - 4*a*c) - sqrt(c*d^2 - b*d*e + a*e^2)*(c*d*g - c*f*e)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2 + 4
*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x + (
b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2)))
/(c^2*d^2*e - b*c*d*e^2 + a*c*e^3), -1/2*(2*sqrt(-c*d^2 + b*d*e - a*e^2)*(c*d*g - c*f*e)*arctan(-1/2*sqrt(-c*d
^2 + b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 +
(a*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) - (c*d^2*g - b*d*g*e + a*g*e^2)*sqrt(c)*log(-
8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c))/(c^2*d^2*e - b*c*d*e^2 + a*c
*e^3), -1/2*(2*(c*d^2*g - b*d*g*e + a*g*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c
^2*x^2 + b*c*x + a*c)) + sqrt(c*d^2 - b*d*e + a*e^2)*(c*d*g - c*f*e)*log(-(8*c^2*d^2*x^2 + 8*b*c*d^2*x + (b^2
+ 4*a*c)*d^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a) + (8*a*b*x
+ (b^2 + 4*a*c)*x^2 + 8*a^2)*e^2 - 2*(4*b*c*d*x^2 + 4*a*b*d + (3*b^2 + 4*a*c)*d*x)*e)/(x^2*e^2 + 2*d*x*e + d^2
)))/(c^2*d^2*e - b*c*d*e^2 + a*c*e^3), -(sqrt(-c*d^2 + b*d*e - a*e^2)*(c*d*g - c*f*e)*arctan(-1/2*sqrt(-c*d^2
+ b*d*e - a*e^2)*(2*c*d*x + b*d - (b*x + 2*a)*e)*sqrt(c*x^2 + b*x + a)/(c^2*d^2*x^2 + b*c*d^2*x + a*c*d^2 + (a
*c*x^2 + a*b*x + a^2)*e^2 - (b*c*d*x^2 + b^2*d*x + a*b*d)*e)) + (c*d^2*g - b*d*g*e + a*g*e^2)*sqrt(-c)*arctan(
1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + a*c)))/(c^2*d^2*e - b*c*d*e^2 + a*c*e^3)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f + g x}{\left (d + e x\right ) \sqrt {a + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((f + g*x)/((d + e*x)*sqrt(a + b*x + c*x**2)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument Type

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {f+g\,x}{\left (d+e\,x\right )\,\sqrt {c\,x^2+b\,x+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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