3.638 \(\int \frac{\sqrt{c+d x}}{1+x^2} \, dx\)

Optimal. Leaf size=316 \[ \frac{d \log \left (-\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right )}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}-\frac{d \log \left (\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right )}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c^2+d^2}+c}-\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right )}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c^2+d^2}+c}+\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right )}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}} \]

[Out]

(d*ArcTanh[(Sqrt[c + Sqrt[c^2 + d^2]] - Sqrt[2]*Sqrt[c + d*x])/Sqrt[c - Sqrt[c^2
+ d^2]]])/(Sqrt[2]*Sqrt[c - Sqrt[c^2 + d^2]]) - (d*ArcTanh[(Sqrt[c + Sqrt[c^2 +
d^2]] + Sqrt[2]*Sqrt[c + d*x])/Sqrt[c - Sqrt[c^2 + d^2]]])/(Sqrt[2]*Sqrt[c - Sq
rt[c^2 + d^2]]) + (d*Log[c + Sqrt[c^2 + d^2] + d*x - Sqrt[2]*Sqrt[c + Sqrt[c^2 +
d^2]]*Sqrt[c + d*x]])/(2*Sqrt[2]*Sqrt[c + Sqrt[c^2 + d^2]]) - (d*Log[c + Sqrt[c
^2 + d^2] + d*x + Sqrt[2]*Sqrt[c + Sqrt[c^2 + d^2]]*Sqrt[c + d*x]])/(2*Sqrt[2]*S
qrt[c + Sqrt[c^2 + d^2]])

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Rubi [A]  time = 0.761056, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{d \log \left (-\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right )}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}-\frac{d \log \left (\sqrt{2} \sqrt{\sqrt{c^2+d^2}+c} \sqrt{c+d x}+\sqrt{c^2+d^2}+c+d x\right )}{2 \sqrt{2} \sqrt{\sqrt{c^2+d^2}+c}}+\frac{d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c^2+d^2}+c}-\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right )}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}}-\frac{d \tanh ^{-1}\left (\frac{\sqrt{\sqrt{c^2+d^2}+c}+\sqrt{2} \sqrt{c+d x}}{\sqrt{c-\sqrt{c^2+d^2}}}\right )}{\sqrt{2} \sqrt{c-\sqrt{c^2+d^2}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x]/(1 + x^2),x]

[Out]

(d*ArcTanh[(Sqrt[c + Sqrt[c^2 + d^2]] - Sqrt[2]*Sqrt[c + d*x])/Sqrt[c - Sqrt[c^2
+ d^2]]])/(Sqrt[2]*Sqrt[c - Sqrt[c^2 + d^2]]) - (d*ArcTanh[(Sqrt[c + Sqrt[c^2 +
d^2]] + Sqrt[2]*Sqrt[c + d*x])/Sqrt[c - Sqrt[c^2 + d^2]]])/(Sqrt[2]*Sqrt[c - Sq
rt[c^2 + d^2]]) + (d*Log[c + Sqrt[c^2 + d^2] + d*x - Sqrt[2]*Sqrt[c + Sqrt[c^2 +
d^2]]*Sqrt[c + d*x]])/(2*Sqrt[2]*Sqrt[c + Sqrt[c^2 + d^2]]) - (d*Log[c + Sqrt[c
^2 + d^2] + d*x + Sqrt[2]*Sqrt[c + Sqrt[c^2 + d^2]]*Sqrt[c + d*x]])/(2*Sqrt[2]*S
qrt[c + Sqrt[c^2 + d^2]])

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Rubi in Sympy [A]  time = 59.4648, size = 291, normalized size = 0.92 \[ \frac{\sqrt{2} d \log{\left (c + d x - \sqrt{2} \sqrt{c + d x} \sqrt{c + \sqrt{c^{2} + d^{2}}} + \sqrt{c^{2} + d^{2}} \right )}}{4 \sqrt{c + \sqrt{c^{2} + d^{2}}}} - \frac{\sqrt{2} d \log{\left (c + d x + \sqrt{2} \sqrt{c + d x} \sqrt{c + \sqrt{c^{2} + d^{2}}} + \sqrt{c^{2} + d^{2}} \right )}}{4 \sqrt{c + \sqrt{c^{2} + d^{2}}}} - \frac{\sqrt{2} d \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{c + d x} - \frac{\sqrt{2} \sqrt{c + \sqrt{c^{2} + d^{2}}}}{2}\right )}{\sqrt{c - \sqrt{c^{2} + d^{2}}}} \right )}}{2 \sqrt{c - \sqrt{c^{2} + d^{2}}}} - \frac{\sqrt{2} d \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt{c + d x} + \frac{\sqrt{2} \sqrt{c + \sqrt{c^{2} + d^{2}}}}{2}\right )}{\sqrt{c - \sqrt{c^{2} + d^{2}}}} \right )}}{2 \sqrt{c - \sqrt{c^{2} + d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(1/2)/(x**2+1),x)

[Out]

sqrt(2)*d*log(c + d*x - sqrt(2)*sqrt(c + d*x)*sqrt(c + sqrt(c**2 + d**2)) + sqrt
(c**2 + d**2))/(4*sqrt(c + sqrt(c**2 + d**2))) - sqrt(2)*d*log(c + d*x + sqrt(2)
*sqrt(c + d*x)*sqrt(c + sqrt(c**2 + d**2)) + sqrt(c**2 + d**2))/(4*sqrt(c + sqrt
(c**2 + d**2))) - sqrt(2)*d*atanh(sqrt(2)*(sqrt(c + d*x) - sqrt(2)*sqrt(c + sqrt
(c**2 + d**2))/2)/sqrt(c - sqrt(c**2 + d**2)))/(2*sqrt(c - sqrt(c**2 + d**2))) -
sqrt(2)*d*atanh(sqrt(2)*(sqrt(c + d*x) + sqrt(2)*sqrt(c + sqrt(c**2 + d**2))/2)
/sqrt(c - sqrt(c**2 + d**2)))/(2*sqrt(c - sqrt(c**2 + d**2)))

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Mathematica [C]  time = 0.0403332, size = 75, normalized size = 0.24 \[ i \sqrt{c+i d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c+i d}}\right )-i \sqrt{c-i d} \tanh ^{-1}\left (\frac{\sqrt{c+d x}}{\sqrt{c-i d}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x]/(1 + x^2),x]

[Out]

(-I)*Sqrt[c - I*d]*ArcTanh[Sqrt[c + d*x]/Sqrt[c - I*d]] + I*Sqrt[c + I*d]*ArcTan
h[Sqrt[c + d*x]/Sqrt[c + I*d]]

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Maple [B]  time = 0.116, size = 570, normalized size = 1.8 \[{\frac{c}{4\,d}\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\ln \left ( dx+c+\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{dx+c}+\sqrt{{c}^{2}+{d}^{2}} \right ) }-{\frac{{c}^{2}}{d}\arctan \left ({1 \left ( 2\,\sqrt{dx+c}+\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}}-{\frac{1}{4\,d}\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{{c}^{2}+{d}^{2}}\ln \left ( dx+c+\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{dx+c}+\sqrt{{c}^{2}+{d}^{2}} \right ) }+{\frac{{c}^{2}+{d}^{2}}{d}\arctan \left ({1 \left ( 2\,\sqrt{dx+c}+\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}}-{\frac{c}{4\,d}\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\ln \left ( \sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{dx+c}-dx-c-\sqrt{{c}^{2}+{d}^{2}} \right ) }+{\frac{{c}^{2}}{d}\arctan \left ({1 \left ( \sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}-2\,\sqrt{dx+c} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}}+{\frac{1}{4\,d}\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{{c}^{2}+{d}^{2}}\ln \left ( \sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}\sqrt{dx+c}-dx-c-\sqrt{{c}^{2}+{d}^{2}} \right ) }-{\frac{{c}^{2}+{d}^{2}}{d}\arctan \left ({1 \left ( \sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}+2\,c}-2\,\sqrt{dx+c} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{c}^{2}+{d}^{2}}-2\,c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(1/2)/(x^2+1),x)

[Out]

1/4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/d*c*ln(d*x+c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(d*
x+c)^(1/2)+(c^2+d^2)^(1/2))-1/d*c^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(d*x
+c)^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))-1/4*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)/d*(c^2+d^2)^(1/2)*ln(d*x+c+(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*(d*x+c)^(1/2)+(c^2+d^2)^(1/2))+1/d*(c^2+d^2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*ar
ctan((2*(d*x+c)^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/
2))-1/4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/d*c*ln((2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(d*x+
c)^(1/2)-d*x-c-(c^2+d^2)^(1/2))+1/d*c^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2
*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(d*x+c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))+1/4*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)/d*(c^2+d^2)^(1/2)*ln((2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*(d*x+c)^(1/2)-d*x-c-(c^2+d^2)^(1/2))-1/d*(c^2+d^2)/(2*(c^2+d^2)^(1/2)-2*c)^(1/2
)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(d*x+c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)
^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x + c}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(x^2 + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x + c)/(x^2 + 1), x)

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Fricas [A]  time = 0.270931, size = 2689, normalized size = 8.51 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(x^2 + 1),x, algorithm="fricas")

[Out]

1/4*sqrt(2)*((c^2 + d^2)^(1/4)*(c - sqrt(c^2 + d^2))*log((8*c^7*d^2 + 16*c^5*d^4
+ 9*c^3*d^6 + c*d^8 + sqrt(2)*(8*c^5*d^3 + 12*c^3*d^5 + 4*c*d^7 - (8*c^4*d^3 +
8*c^2*d^5 + d^7)*sqrt(c^2 + d^2))*(c^2 + d^2)^(1/4)*sqrt(d*x + c)*sqrt((c^2 + d^
2 - sqrt(c^2 + d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c)) + (8*c^6*d^3 + 16*c^
4*d^5 + 9*c^2*d^7 + d^9)*x + (8*c^6*d^2 + 16*c^4*d^4 + 9*c^2*d^6 + d^8 - 4*(2*c^
5*d^2 + 3*c^3*d^4 + c*d^6)*sqrt(c^2 + d^2))*sqrt(c^2 + d^2) - 4*(2*c^6*d^2 + 3*c
^4*d^4 + c^2*d^6 + (2*c^5*d^3 + 3*c^3*d^5 + c*d^7)*x)*sqrt(c^2 + d^2))/(8*c^6 +
16*c^4*d^2 + 9*c^2*d^4 + d^6 - 4*(2*c^5 + 3*c^3*d^2 + c*d^4)*sqrt(c^2 + d^2))) -
(c^2 + d^2)^(1/4)*(c - sqrt(c^2 + d^2))*log((8*c^7*d^2 + 16*c^5*d^4 + 9*c^3*d^6
+ c*d^8 - sqrt(2)*(8*c^5*d^3 + 12*c^3*d^5 + 4*c*d^7 - (8*c^4*d^3 + 8*c^2*d^5 +
d^7)*sqrt(c^2 + d^2))*(c^2 + d^2)^(1/4)*sqrt(d*x + c)*sqrt((c^2 + d^2 - sqrt(c^2
+ d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c)) + (8*c^6*d^3 + 16*c^4*d^5 + 9*c^
2*d^7 + d^9)*x + (8*c^6*d^2 + 16*c^4*d^4 + 9*c^2*d^6 + d^8 - 4*(2*c^5*d^2 + 3*c^
3*d^4 + c*d^6)*sqrt(c^2 + d^2))*sqrt(c^2 + d^2) - 4*(2*c^6*d^2 + 3*c^4*d^4 + c^2
*d^6 + (2*c^5*d^3 + 3*c^3*d^5 + c*d^7)*x)*sqrt(c^2 + d^2))/(8*c^6 + 16*c^4*d^2 +
9*c^2*d^4 + d^6 - 4*(2*c^5 + 3*c^3*d^2 + c*d^4)*sqrt(c^2 + d^2))) + 4*(c^2 + d^
2)^(1/4)*sqrt(d^2)*arctan((c^2 + d^2)^(1/4)*(c*sqrt(d^2) - sqrt(c^2 + d^2)*sqrt(
d^2))/(sqrt(2)*(c*d - sqrt(c^2 + d^2)*d)*sqrt(d*x + c)*sqrt((c^2 + d^2 - sqrt(c^
2 + d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c)) + sqrt(2)*(c - sqrt(c^2 + d^2))
*sqrt((8*c^7*d^2 + 16*c^5*d^4 + 9*c^3*d^6 + c*d^8 + sqrt(2)*(8*c^5*d^3 + 12*c^3*
d^5 + 4*c*d^7 - (8*c^4*d^3 + 8*c^2*d^5 + d^7)*sqrt(c^2 + d^2))*(c^2 + d^2)^(1/4)
*sqrt(d*x + c)*sqrt((c^2 + d^2 - sqrt(c^2 + d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 +
d^2)*c)) + (8*c^6*d^3 + 16*c^4*d^5 + 9*c^2*d^7 + d^9)*x + (8*c^6*d^2 + 16*c^4*d^
4 + 9*c^2*d^6 + d^8 - 4*(2*c^5*d^2 + 3*c^3*d^4 + c*d^6)*sqrt(c^2 + d^2))*sqrt(c^
2 + d^2) - 4*(2*c^6*d^2 + 3*c^4*d^4 + c^2*d^6 + (2*c^5*d^3 + 3*c^3*d^5 + c*d^7)*
x)*sqrt(c^2 + d^2))/(8*c^6 + 16*c^4*d^2 + 9*c^2*d^4 + d^6 - 4*(2*c^5 + 3*c^3*d^2
+ c*d^4)*sqrt(c^2 + d^2)))*sqrt((c^2 + d^2 - sqrt(c^2 + d^2)*c)/(2*c^2 + d^2 -
2*sqrt(c^2 + d^2)*c)) + (c^2 + d^2)^(1/4)*d^2)) + 4*(c^2 + d^2)^(1/4)*sqrt(d^2)*
arctan((c^2 + d^2)^(1/4)*(c*sqrt(d^2) - sqrt(c^2 + d^2)*sqrt(d^2))/(sqrt(2)*(c*d
- sqrt(c^2 + d^2)*d)*sqrt(d*x + c)*sqrt((c^2 + d^2 - sqrt(c^2 + d^2)*c)/(2*c^2
+ d^2 - 2*sqrt(c^2 + d^2)*c)) + sqrt(2)*(c - sqrt(c^2 + d^2))*sqrt((8*c^7*d^2 +
16*c^5*d^4 + 9*c^3*d^6 + c*d^8 - sqrt(2)*(8*c^5*d^3 + 12*c^3*d^5 + 4*c*d^7 - (8*
c^4*d^3 + 8*c^2*d^5 + d^7)*sqrt(c^2 + d^2))*(c^2 + d^2)^(1/4)*sqrt(d*x + c)*sqrt
((c^2 + d^2 - sqrt(c^2 + d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c)) + (8*c^6*d
^3 + 16*c^4*d^5 + 9*c^2*d^7 + d^9)*x + (8*c^6*d^2 + 16*c^4*d^4 + 9*c^2*d^6 + d^8
- 4*(2*c^5*d^2 + 3*c^3*d^4 + c*d^6)*sqrt(c^2 + d^2))*sqrt(c^2 + d^2) - 4*(2*c^6
*d^2 + 3*c^4*d^4 + c^2*d^6 + (2*c^5*d^3 + 3*c^3*d^5 + c*d^7)*x)*sqrt(c^2 + d^2))
/(8*c^6 + 16*c^4*d^2 + 9*c^2*d^4 + d^6 - 4*(2*c^5 + 3*c^3*d^2 + c*d^4)*sqrt(c^2
+ d^2)))*sqrt((c^2 + d^2 - sqrt(c^2 + d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c
)) - (c^2 + d^2)^(1/4)*d^2)))/((c - sqrt(c^2 + d^2))*sqrt((c^2 + d^2 - sqrt(c^2
+ d^2)*c)/(2*c^2 + d^2 - 2*sqrt(c^2 + d^2)*c)))

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Sympy [A]  time = 10.6863, size = 53, normalized size = 0.17 \[ 2 d \operatorname{RootSum}{\left (256 t^{4} d^{4} + 32 t^{2} c d^{2} + c^{2} + d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} d^{2} + 4 t c + \sqrt{c + d x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(1/2)/(x**2+1),x)

[Out]

2*d*RootSum(256*_t**4*d**4 + 32*_t**2*c*d**2 + c**2 + d**2, Lambda(_t, _t*log(64
*_t**3*d**2 + 4*_t*c + sqrt(c + d*x))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x + c}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x + c)/(x^2 + 1),x, algorithm="giac")

[Out]

integrate(sqrt(d*x + c)/(x^2 + 1), x)