3.4.17 \(\int \frac {d+e x}{\sqrt {b x+c x^2}} \, dx\) [317]

Optimal. Leaf size=55 \[ \frac {e \sqrt {b x+c x^2}}{c}+\frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {654, 634, 212} \begin {gather*} \frac {(2 c d-b e) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}+\frac {e \sqrt {b x+c x^2}}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/Sqrt[b*x + c*x^2],x]

[Out]

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

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

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Mathematica [A]
time = 0.07, size = 77, normalized size = 1.40 \begin {gather*} \frac {\sqrt {c} e x (b+c x)+(-2 c d+b e) \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{c^{3/2} \sqrt {x (b+c x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c]*e*x*(b + c*x) + (-2*c*d + b*e)*Sqrt[x]*Sqrt[b + c*x]*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x]])/(c^(3/2
)*Sqrt[x*(b + c*x)])

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Maple [A]
time = 0.40, size = 79, normalized size = 1.44

method	result	size
default	\(e \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )+\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}\)	\(79\)
risch	\(\frac {e x \left (c x +b \right )}{c \sqrt {x \left (c x +b \right )}}-\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right ) b e}{2 c^{\frac {3}{2}}}+\frac {d \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*((c*x^2+b*x)^(1/2)/c-1/2*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2)))+d*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x)^(1/2))/c^(1/2)

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Maxima [A]
time = 0.27, size = 77, normalized size = 1.40 \begin {gather*} \frac {d \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} - \frac {b e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{2} + b x} e}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 1.57, size = 123, normalized size = 2.24 \begin {gather*} \left [\frac {2 \, \sqrt {c x^{2} + b x} c e - {\left (2 \, c d - b e\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{2}}, -\frac {{\left (2 \, c d - b e\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - \sqrt {c x^{2} + b x} c e}{c^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*sqrt(c*x^2 + b*x)*c*e - (2*c*d - b*e)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)))/c^2, -((2*
c*d - b*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - sqrt(c*x^2 + b*x)*c*e)/c^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((d + e*x)/sqrt(x*(b + c*x)), x)

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Giac [A]
time = 1.21, size = 63, normalized size = 1.15 \begin {gather*} \frac {\sqrt {c x^{2} + b x} e}{c} - \frac {{\left (2 \, c d - b e\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{2 \, c^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c*x^2 + b*x)*e/c - 1/2*(2*c*d - b*e)*log(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(3/2)

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Mupad [B]
time = 0.51, size = 77, normalized size = 1.40 \begin {gather*} \frac {e\,\sqrt {c\,x^2+b\,x}}{c}+\frac {d\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{\sqrt {c}}-\frac {b\,e\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{2\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2)^(1/2),x)

[Out]

(e*(b*x + c*x^2)^(1/2))/c + (d*log((b/2 + c*x)/c^(1/2) + (b*x + c*x^2)^(1/2)))/c^(1/2) - (b*e*log((b/2 + c*x)/
c^(1/2) + (b*x + c*x^2)^(1/2)))/(2*c^(3/2))

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