3.17.28 \(\int \frac {1-a x+b \sqrt {a+b x}}{\sqrt {a+b x} (x+a b \sqrt {a+b x})} \, dx\)

Optimal. Leaf size=110 \[ \frac {2 \left (a^3 b^3+a b^3-2\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2 b+b\right ) \log \left (a b^2 \sqrt {a+b x}+b x\right )-\frac {2 a \sqrt {a+b x}}{b} \]

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Rubi [A]  time = 0.56, antiderivative size = 104, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {1984, 1657, 634, 618, 206, 628} \begin {gather*} -\frac {2 \left (2-a \left (a^2+1\right ) b^3\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2+1\right ) b \log \left (a b \sqrt {a+b x}+x\right )-\frac {2 a \sqrt {a+b x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - a*x + b*Sqrt[a + b*x])/(Sqrt[a + b*x]*(x + a*b*Sqrt[a + b*x])),x]

[Out]

(-2*a*Sqrt[a + b*x])/b - (2*(2 - a*(1 + a^2)*b^3)*ArcTanh[(a*b^2 + 2*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[4 + a*b^4])]
)/(Sqrt[a]*Sqrt[4 + a*b^4]) + (1 + a^2)*b*Log[x + a*b*Sqrt[a + b*x]]

Rule 206

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

Rule 618

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

Rule 628

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

Rule 634

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {align*} \int \frac {1-a x+b \sqrt {a+b x}}{\sqrt {a+b x} \left (x+a b \sqrt {a+b x}\right )} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {a^2+b+b^2 x-a x^2}{x^2+a \left (-1+b^2 x\right )} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \frac {a^2+b+b^2 x-a x^2}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=\frac {2 \operatorname {Subst}\left (\int \left (-a+\frac {b+\left (1+a^2\right ) b^2 x}{-a+a b^2 x+x^2}\right ) \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\frac {2 \operatorname {Subst}\left (\int \frac {b+\left (1+a^2\right ) b^2 x}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )}{b}\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\left (\left (1+a^2\right ) b\right ) \operatorname {Subst}\left (\int \frac {a b^2+2 x}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )+\left (2-a \left (1+a^2\right ) b^3\right ) \operatorname {Subst}\left (\int \frac {1}{-a+a b^2 x+x^2} \, dx,x,\sqrt {a+b x}\right )\\ &=-\frac {2 a \sqrt {a+b x}}{b}+\left (1+a^2\right ) b \log \left (x+a b \sqrt {a+b x}\right )-\left (2 \left (2-a \left (1+a^2\right ) b^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \left (4+a b^4\right )-x^2} \, dx,x,a b^2+2 \sqrt {a+b x}\right )\\ &=-\frac {2 a \sqrt {a+b x}}{b}-\frac {2 \left (2-a \left (1+a^2\right ) b^3\right ) \tanh ^{-1}\left (\frac {a b^2+2 \sqrt {a+b x}}{\sqrt {a} \sqrt {4+a b^4}}\right )}{\sqrt {a} \sqrt {4+a b^4}}+\left (1+a^2\right ) b \log \left (x+a b \sqrt {a+b x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 106, normalized size = 0.96 \begin {gather*} -\frac {2 \left (a^3 b^3+a b^3-2\right ) \tanh ^{-1}\left (\frac {-a b^2-2 \sqrt {a+b x}}{\sqrt {a} \sqrt {a b^4+4}}\right )}{\sqrt {a} \sqrt {a b^4+4}}+\left (a^2+1\right ) b \log \left (a b \sqrt {a+b x}+x\right )-\frac {2 a \sqrt {a+b x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - a*x + b*Sqrt[a + b*x])/(Sqrt[a + b*x]*(x + a*b*Sqrt[a + b*x])),x]

[Out]

(-2*a*Sqrt[a + b*x])/b - (2*(-2 + a*b^3 + a^3*b^3)*ArcTanh[(-(a*b^2) - 2*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[4 + a*b^
4])])/(Sqrt[a]*Sqrt[4 + a*b^4]) + (1 + a^2)*b*Log[x + a*b*Sqrt[a + b*x]]

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IntegrateAlgebraic [A]  time = 0.16, size = 124, normalized size = 1.13 \begin {gather*} -\frac {2 a \sqrt {a+b x}}{b}+\frac {2 \left (-2+a b^3+a^3 b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a} b^2}{\sqrt {4+a b^4}}+\frac {2 \sqrt {a+b x}}{\sqrt {a} \sqrt {4+a b^4}}\right )}{\sqrt {a} \sqrt {4+a b^4}}+\left (b+a^2 b\right ) \log \left (b x+a b^2 \sqrt {a+b x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - a*x + b*Sqrt[a + b*x])/(Sqrt[a + b*x]*(x + a*b*Sqrt[a + b*x])),x]

[Out]

(-2*a*Sqrt[a + b*x])/b + (2*(-2 + a*b^3 + a^3*b^3)*ArcTanh[(Sqrt[a]*b^2)/Sqrt[4 + a*b^4] + (2*Sqrt[a + b*x])/(
Sqrt[a]*Sqrt[4 + a*b^4])])/(Sqrt[a]*Sqrt[4 + a*b^4]) + (b + a^2*b)*Log[b*x + a*b^2*Sqrt[a + b*x]]

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fricas [A]  time = 0.50, size = 392, normalized size = 3.56 \begin {gather*} \left [\frac {\sqrt {a^{2} b^{4} + 4 \, a} {\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \log \left (\frac {2 \, a^{3} b^{3} - 2 \, b x^{2} + {\left (a^{2} b^{4} - 4 \, a\right )} x + \sqrt {a^{2} b^{4} + 4 \, a} {\left (a b^{2} x + 2 \, a^{2} b\right )} + {\left (a^{3} b^{5} + 4 \, a^{2} b + \sqrt {a^{2} b^{4} + 4 \, a} {\left (a^{2} b^{3} - 2 \, x\right )}\right )} \sqrt {b x + a}}{a^{2} b^{3} x + a^{3} b^{2} - x^{2}}\right ) + {\left ({\left (a^{4} + a^{2}\right )} b^{6} + 4 \, {\left (a^{3} + a\right )} b^{2}\right )} \log \left (\sqrt {b x + a} a b + x\right ) - 2 \, {\left (a^{3} b^{4} + 4 \, a^{2}\right )} \sqrt {b x + a}}{a^{2} b^{5} + 4 \, a b}, -\frac {2 \, \sqrt {-a^{2} b^{4} - 4 \, a} {\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \arctan \left (\frac {\sqrt {-a^{2} b^{4} - 4 \, a} a b^{2} + 2 \, \sqrt {-a^{2} b^{4} - 4 \, a} \sqrt {b x + a}}{a^{2} b^{4} + 4 \, a}\right ) - {\left ({\left (a^{4} + a^{2}\right )} b^{6} + 4 \, {\left (a^{3} + a\right )} b^{2}\right )} \log \left (\sqrt {b x + a} a b + x\right ) + 2 \, {\left (a^{3} b^{4} + 4 \, a^{2}\right )} \sqrt {b x + a}}{a^{2} b^{5} + 4 \, a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a*x+b*(b*x+a)^(1/2))/(b*x+a)^(1/2)/(x+a*b*(b*x+a)^(1/2)),x, algorithm="fricas")

[Out]

[(sqrt(a^2*b^4 + 4*a)*((a^3 + a)*b^4 - 2*b)*log((2*a^3*b^3 - 2*b*x^2 + (a^2*b^4 - 4*a)*x + sqrt(a^2*b^4 + 4*a)
*(a*b^2*x + 2*a^2*b) + (a^3*b^5 + 4*a^2*b + sqrt(a^2*b^4 + 4*a)*(a^2*b^3 - 2*x))*sqrt(b*x + a))/(a^2*b^3*x + a
^3*b^2 - x^2)) + ((a^4 + a^2)*b^6 + 4*(a^3 + a)*b^2)*log(sqrt(b*x + a)*a*b + x) - 2*(a^3*b^4 + 4*a^2)*sqrt(b*x
+ a))/(a^2*b^5 + 4*a*b), -(2*sqrt(-a^2*b^4 - 4*a)*((a^3 + a)*b^4 - 2*b)*arctan((sqrt(-a^2*b^4 - 4*a)*a*b^2 +
2*sqrt(-a^2*b^4 - 4*a)*sqrt(b*x + a))/(a^2*b^4 + 4*a)) - ((a^4 + a^2)*b^6 + 4*(a^3 + a)*b^2)*log(sqrt(b*x + a)
*a*b + x) + 2*(a^3*b^4 + 4*a^2)*sqrt(b*x + a))/(a^2*b^5 + 4*a*b)]

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giac [A]  time = 0.36, size = 100, normalized size = 0.91 \begin {gather*} {\left (a^{2} b + b\right )} \log \left (\sqrt {b x + a} a b^{2} + b x\right ) - \frac {2 \, {\left (a^{3} b^{3} + a b^{3} - 2\right )} \arctan \left (\frac {a b^{2} + 2 \, \sqrt {b x + a}}{\sqrt {-a^{2} b^{4} - 4 \, a}}\right )}{\sqrt {-a^{2} b^{4} - 4 \, a}} - \frac {2 \, \sqrt {b x + a} a}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a*x+b*(b*x+a)^(1/2))/(b*x+a)^(1/2)/(x+a*b*(b*x+a)^(1/2)),x, algorithm="giac")

[Out]

(a^2*b + b)*log(sqrt(b*x + a)*a*b^2 + b*x) - 2*(a^3*b^3 + a*b^3 - 2)*arctan((a*b^2 + 2*sqrt(b*x + a))/sqrt(-a^
2*b^4 - 4*a))/sqrt(-a^2*b^4 - 4*a) - 2*sqrt(b*x + a)*a/b

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maple [A]  time = 0.12, size = 106, normalized size = 0.96

method	result	size
derivativedivides	\(-\frac {2 \left (a \sqrt {b x +a}-b \left (\frac {\left (a^{2} b +b \right ) \ln \left (b x +a \,b^{2} \sqrt {b x +a}\right )}{2}-\frac {2 \left (1-\frac {\left (a^{2} b +b \right ) a \,b^{2}}{2}\right ) \arctanh \left (\frac {a \,b^{2}+2 \sqrt {b x +a}}{\sqrt {a^{2} b^{4}+4 a}}\right )}{\sqrt {a^{2} b^{4}+4 a}}\right )\right )}{b}\)	\(106\)
default	\(-\frac {2 \left (a \sqrt {b x +a}-b \left (\frac {\left (a^{2} b +b \right ) \ln \left (b x +a \,b^{2} \sqrt {b x +a}\right )}{2}-\frac {2 \left (1-\frac {\left (a^{2} b +b \right ) a \,b^{2}}{2}\right ) \arctanh \left (\frac {a \,b^{2}+2 \sqrt {b x +a}}{\sqrt {a^{2} b^{4}+4 a}}\right )}{\sqrt {a^{2} b^{4}+4 a}}\right )\right )}{b}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-a*x+b*(b*x+a)^(1/2))/(b*x+a)^(1/2)/(x+a*b*(b*x+a)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

-2/b*(a*(b*x+a)^(1/2)-b*(1/2*(a^2*b+b)*ln(b*x+a*b^2*(b*x+a)^(1/2))-2*(1-1/2*(a^2*b+b)*a*b^2)/(a^2*b^4+4*a)^(1/
2)*arctanh((a*b^2+2*(b*x+a)^(1/2))/(a^2*b^4+4*a)^(1/2))))

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maxima [A]  time = 0.31, size = 125, normalized size = 1.14 \begin {gather*} \frac {{\left (a^{2} + 1\right )} b^{2} \log \left (\sqrt {b x + a} a b^{2} + b x\right ) - 2 \, \sqrt {b x + a} a - \frac {{\left ({\left (a^{3} + a\right )} b^{4} - 2 \, b\right )} \log \left (\frac {a b^{2} - \sqrt {{\left (a b^{4} + 4\right )} a} + 2 \, \sqrt {b x + a}}{a b^{2} + \sqrt {{\left (a b^{4} + 4\right )} a} + 2 \, \sqrt {b x + a}}\right )}{\sqrt {{\left (a b^{4} + 4\right )} a}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a*x+b*(b*x+a)^(1/2))/(b*x+a)^(1/2)/(x+a*b*(b*x+a)^(1/2)),x, algorithm="maxima")

[Out]

((a^2 + 1)*b^2*log(sqrt(b*x + a)*a*b^2 + b*x) - 2*sqrt(b*x + a)*a - ((a^3 + a)*b^4 - 2*b)*log((a*b^2 - sqrt((a
*b^4 + 4)*a) + 2*sqrt(b*x + a))/(a*b^2 + sqrt((a*b^4 + 4)*a) + 2*sqrt(b*x + a)))/sqrt((a*b^4 + 4)*a))/b

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mupad [B]  time = 1.19, size = 279, normalized size = 2.54 \begin {gather*} \frac {\ln \left (4\,a+2\,\sqrt {a\,\left (a\,b^4+4\right )}\,\sqrt {a+b\,x}+a^2\,b^4+a\,b^2\,\sqrt {a\,\left (a\,b^4+4\right )}\right )\,\left (a\,b\,\left (a\,b^4+4\right )-2\,\sqrt {a\,\left (a\,b^4+4\right )}+a^3\,b\,\left (a\,b^4+4\right )+a\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}+a^3\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}\right )}{a\,\left (a\,b^4+4\right )}-\frac {2\,a\,\sqrt {a+b\,x}}{b}+\frac {\ln \left (4\,a-2\,\sqrt {a\,\left (a\,b^4+4\right )}\,\sqrt {a+b\,x}+a^2\,b^4-a\,b^2\,\sqrt {a\,\left (a\,b^4+4\right )}\right )\,\left (2\,\sqrt {a\,\left (a\,b^4+4\right )}+a\,b\,\left (a\,b^4+4\right )+a^3\,b\,\left (a\,b^4+4\right )-a\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}-a^3\,b^3\,\sqrt {a\,\left (a\,b^4+4\right )}\right )}{a\,\left (a\,b^4+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*(a + b*x)^(1/2) - a*x + 1)/((x + a*b*(a + b*x)^(1/2))*(a + b*x)^(1/2)),x)

[Out]

(log(4*a + 2*(a*(a*b^4 + 4))^(1/2)*(a + b*x)^(1/2) + a^2*b^4 + a*b^2*(a*(a*b^4 + 4))^(1/2))*(a*b*(a*b^4 + 4) -
2*(a*(a*b^4 + 4))^(1/2) + a^3*b*(a*b^4 + 4) + a*b^3*(a*(a*b^4 + 4))^(1/2) + a^3*b^3*(a*(a*b^4 + 4))^(1/2)))/(
a*(a*b^4 + 4)) - (2*a*(a + b*x)^(1/2))/b + (log(4*a - 2*(a*(a*b^4 + 4))^(1/2)*(a + b*x)^(1/2) + a^2*b^4 - a*b^
2*(a*(a*b^4 + 4))^(1/2))*(2*(a*(a*b^4 + 4))^(1/2) + a*b*(a*b^4 + 4) + a^3*b*(a*b^4 + 4) - a*b^3*(a*(a*b^4 + 4)
)^(1/2) - a^3*b^3*(a*(a*b^4 + 4))^(1/2)))/(a*(a*b^4 + 4))

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sympy [A]  time = 92.60, size = 144, normalized size = 1.31 \begin {gather*} - \frac {2 a \sqrt {a + b x}}{b} - 2 b \left (a^{2} + 1\right ) \log {\left (\frac {1}{\sqrt {a + b x}} \right )} - \frac {\left (- a^{3} b - a b\right ) \log {\left (\frac {a b^{2}}{\sqrt {a + b x}} - \frac {a}{a + b x} + 1 \right )}}{a} - \frac {4 \left (a^{3} b^{3} + a b^{3} + \frac {b^{2} \left (- a^{3} b - a b\right )}{2} - 1\right ) \operatorname {atan}{\left (\frac {2 \left (- \frac {b^{2}}{2} + \frac {1}{\sqrt {a + b x}}\right )}{\sqrt {- \frac {a b^{4} + 4}{a}}} \right )}}{a \sqrt {- \frac {a b^{4} + 4}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a*x+b*(b*x+a)**(1/2))/(b*x+a)**(1/2)/(x+a*b*(b*x+a)**(1/2)),x)

[Out]

-2*a*sqrt(a + b*x)/b - 2*b*(a**2 + 1)*log(1/sqrt(a + b*x)) - (-a**3*b - a*b)*log(a*b**2/sqrt(a + b*x) - a/(a +
b*x) + 1)/a - 4*(a**3*b**3 + a*b**3 + b**2*(-a**3*b - a*b)/2 - 1)*atan(2*(-b**2/2 + 1/sqrt(a + b*x))/sqrt(-(a
*b**4 + 4)/a))/(a*sqrt(-(a*b**4 + 4)/a))

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