3.609 \(\int \frac{\sqrt{d+e x}}{a+c x^2} \, dx\)

Optimal. Leaf size=478 \[ \frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[Out]

(e*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d - Sqr
t[c*d^2 + a*e^2]]) - (e*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]
*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(3/4)
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c
^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])
/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - (e*Log[Sqrt[c*d^2 +
a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 1.10981, antiderivative size = 478, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}-\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} c^{3/4} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} c^{3/4} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(e*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d - Sqr
t[c*d^2 + a*e^2]]) - (e*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]
*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*c^(3/4)
*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) + (e*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c
^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])
/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) - (e*Log[Sqrt[c*d^2 +
a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] +
Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*c^(3/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi in Sympy [A]  time = 119.763, size = 430, normalized size = 0.9 \[ \frac{\sqrt{2} e \log{\left (d + e x + \frac{\sqrt{a e^{2} + c d^{2}}}{\sqrt{c}} - \frac{\sqrt{2} \sqrt{d + e x} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}} - \frac{\sqrt{2} e \log{\left (d + e x + \frac{\sqrt{a e^{2} + c d^{2}}}{\sqrt{c}} + \frac{\sqrt{2} \sqrt{d + e x} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}}{\sqrt [4]{c}} \right )}}{4 c^{\frac{3}{4}} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}} - \frac{\sqrt{2} e \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{c} \sqrt{d + e x} - \frac{\sqrt{2 \sqrt{c} d + 2 \sqrt{a e^{2} + c d^{2}}}}{2}\right )}{\sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \right )}}{2 c^{\frac{3}{4}} \sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} - \frac{\sqrt{2} e \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{c} \sqrt{d + e x} + \frac{\sqrt{2 \sqrt{c} d + 2 \sqrt{a e^{2} + c d^{2}}}}{2}\right )}{\sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \right )}}{2 c^{\frac{3}{4}} \sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*e*log(d + e*x + sqrt(a*e**2 + c*d**2)/sqrt(c) - sqrt(2)*sqrt(d + e*x)*sq
rt(sqrt(c)*d + sqrt(a*e**2 + c*d**2))/c**(1/4))/(4*c**(3/4)*sqrt(sqrt(c)*d + sqr
t(a*e**2 + c*d**2))) - sqrt(2)*e*log(d + e*x + sqrt(a*e**2 + c*d**2)/sqrt(c) + s
qrt(2)*sqrt(d + e*x)*sqrt(sqrt(c)*d + sqrt(a*e**2 + c*d**2))/c**(1/4))/(4*c**(3/
4)*sqrt(sqrt(c)*d + sqrt(a*e**2 + c*d**2))) - sqrt(2)*e*atanh(sqrt(2)*(c**(1/4)*
sqrt(d + e*x) - sqrt(2*sqrt(c)*d + 2*sqrt(a*e**2 + c*d**2))/2)/sqrt(sqrt(c)*d -
sqrt(a*e**2 + c*d**2)))/(2*c**(3/4)*sqrt(sqrt(c)*d - sqrt(a*e**2 + c*d**2))) - s
qrt(2)*e*atanh(sqrt(2)*(c**(1/4)*sqrt(d + e*x) + sqrt(2*sqrt(c)*d + 2*sqrt(a*e**
2 + c*d**2))/2)/sqrt(sqrt(c)*d - sqrt(a*e**2 + c*d**2)))/(2*c**(3/4)*sqrt(sqrt(c
)*d - sqrt(a*e**2 + c*d**2)))

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Mathematica [C]  time = 0.293262, size = 140, normalized size = 0.29 \[ -\frac{i \left (\sqrt{c d-i \sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )-\sqrt{c d+i \sqrt{a} \sqrt{c} e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )\right )}{\sqrt{a} c} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d
- I*Sqrt[a]*Sqrt[c]*e]] - Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]*ArcTanh[(Sqrt[c]*Sqrt[
d + e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]]))/(Sqrt[a]*c)

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Maple [B]  time = 0.072, size = 1176, normalized size = 2.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(1/2)/(c*x^2+a),x)

[Out]

-1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*ln(-(e*x+d)*c^(1/2)+(2*
(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*d+1/4*(2
*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*ln(-(e*x+d)*c^(1/2)+(2*(c*(a*e
^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)-(a*e^2+c*d^2)^(1/2))*(a*c*e^2+c^2*d^
2)^(1/2)+1/2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*(2*(c*(a*e^2+c*
d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2
)-2*c*d)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d
)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*
d-1/2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*(2*(c*(a*e^2+c*d^2))^(
1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d
)^(1/2)*arctan((-2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)
)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1/2))*(a*c*e^
2+c^2*d^2)^(1/2)+1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*ln((e*x
+d)*c^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^
(1/2))*d-1/4*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*ln((e*x+d)*c^(1
/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(e*x+d)^(1/2)+(a*e^2+c*d^2)^(1/2))*(
a*c*e^2+c^2*d^2)^(1/2)-1/2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/e/c^(1/2)*(
2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^
2+c*d^2))^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2)
)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2
*c*d)^(1/2))*d+1/2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)/a/c^(3/2)/e*(2*(c*(a*
e^2+c*d^2))^(1/2)+2*c*d)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2)
)^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)+(2*(c*(a*e^2+c*d^2))^(1/2)+
2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*(c*(a*e^2+c*d^2))^(1/2)-2*c*d)^(1
/2))*(a*c*e^2+c^2*d^2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(e*x + d)/(c*x^2 + a), x)

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Fricas [A]  time = 0.220679, size = 479, normalized size = 1. \[ -\frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (a c^{2} \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \log \left (-a c^{2} \sqrt{-\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} + d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) + \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (a c^{2} \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) - \frac{1}{2} \, \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \log \left (-a c^{2} \sqrt{\frac{a c \sqrt{-\frac{e^{2}}{a c^{3}}} - d}{a c}} \sqrt{-\frac{e^{2}}{a c^{3}}} + \sqrt{e x + d} e\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.8257, size = 75, normalized size = 0.16 \[ 2 e \operatorname{RootSum}{\left (256 t^{4} a^{2} c^{3} e^{4} + 32 t^{2} a c^{2} d e^{2} + a e^{2} + c d^{2}, \left ( t \mapsto t \log{\left (64 t^{3} a c^{2} e^{2} + 4 t c d + \sqrt{d + e x} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 42.8623, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done