3.610 \(\int \frac{1}{\sqrt{d+e x} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=538 \[ -\frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

[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^(1/4)*Sqrt[c*d^2 + a*e^2]*
Sqrt[Sqrt[c]*d - Sqrt[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^(1/4)*Sqrt[c*d^2 + a*e^2]*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^(1/4)*Sqrt[c*d^2 + a*e^2]*
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^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi [A]  time = 1.23106, antiderivative size = 538, 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 \[ -\frac{e \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{2 \sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{\sqrt{2} \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(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^(1/4)*Sqrt[c*d^2 + a*e^2]*
Sqrt[Sqrt[c]*d - Sqrt[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^(1/4)*Sqrt[c*d^2 + a*e^2]*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^(1/4)*Sqrt[c*d^2 + a*e^2]*
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^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

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Rubi in Sympy [A]  time = 163.122, size = 484, normalized size = 0.9 \[ - \frac{\sqrt{2} e \log{\left (d + e x + \frac{\sqrt{a e^{2} + c d^{2}}}{\sqrt{c}} - \frac{\sqrt{2} \sqrt{d + e x} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}}{\sqrt [4]{c}} \right )}}{4 \sqrt [4]{c} \sqrt{a e^{2} + c d^{2}} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}} + \frac{\sqrt{2} e \log{\left (d + e x + \frac{\sqrt{a e^{2} + c d^{2}}}{\sqrt{c}} + \frac{\sqrt{2} \sqrt{d + e x} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}}{\sqrt [4]{c}} \right )}}{4 \sqrt [4]{c} \sqrt{a e^{2} + c d^{2}} \sqrt{\sqrt{c} d + \sqrt{a e^{2} + c d^{2}}}} - \frac{\sqrt{2} e \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{c} \sqrt{d + e x} - \frac{\sqrt{2 \sqrt{c} d + 2 \sqrt{a e^{2} + c d^{2}}}}{2}\right )}{\sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \right )}}{2 \sqrt [4]{c} \sqrt{a e^{2} + c d^{2}} \sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} - \frac{\sqrt{2} e \operatorname{atanh}{\left (\frac{\sqrt{2} \left (\sqrt [4]{c} \sqrt{d + e x} + \frac{\sqrt{2 \sqrt{c} d + 2 \sqrt{a e^{2} + c d^{2}}}}{2}\right )}{\sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \right )}}{2 \sqrt [4]{c} \sqrt{a e^{2} + c d^{2}} \sqrt{\sqrt{c} d - \sqrt{a e^{2} + c d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*e*log(d + e*x + sqrt(a*e**2 + c*d**2)/sqrt(c) - sqrt(2)*sqrt(d + e*x)*s
qrt(sqrt(c)*d + sqrt(a*e**2 + c*d**2))/c**(1/4))/(4*c**(1/4)*sqrt(a*e**2 + c*d**
2)*sqrt(sqrt(c)*d + sqrt(a*e**2 + c*d**2))) + sqrt(2)*e*log(d + e*x + sqrt(a*e**
2 + c*d**2)/sqrt(c) + sqrt(2)*sqrt(d + e*x)*sqrt(sqrt(c)*d + sqrt(a*e**2 + c*d**
2))/c**(1/4))/(4*c**(1/4)*sqrt(a*e**2 + c*d**2)*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**(1/4)
*sqrt(a*e**2 + c*d**2)*sqrt(sqrt(c)*d - sqrt(a*e**2 + c*d**2))) - sqrt(2)*e*atan
h(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**(1/4)*sqrt(a*e**2 + c*d**2)*sq
rt(sqrt(c)*d - sqrt(a*e**2 + c*d**2)))

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Mathematica [C]  time = 0.175649, size = 137, normalized size = 0.25 \[ -\frac{i \left (\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(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]*S
qrt[c]*e]]/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]))/Sqrt[a]

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Maple [B]  time = 0.074, size = 1546, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/(a*e^2+c*d^2)^(1/2)/a/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))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/
2)*d-1/4/(a*e^2+c*d^2)^(1/2)/c/a/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))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c
*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-2*e/(a*e^2+c*d^2)^(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))-1/2/(a*e^2+c*d^2)^(1/2)/a/e/(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))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(
a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+1/2/(a*e^2+c*d^2)^(1/2)/c/a/e/(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))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/
2)*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)-1/4/(a*e^2+c*
d^2)^(1/2)/a/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))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*d+1/4/(a*e^
2+c*d^2)^(1/2)/c/a/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))*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*c*
e^2+c^2*d^2)^(1/2)+2*e/(a*e^2+c*d^2)^(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))+1/2/(a*e^2+c*d^2)^(1/2)/a/e/(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))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2*d^2)^
(1/2)+2*c*d)^(1/2)*d-1/2/(a*e^2+c*d^2)^(1/2)/c/a/e/(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))*(2*(c*(a*e^2+c*d^2))^(1/2)+2*c*d)^(1/2)*(2*(a*c*e^2+c^2
*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. \[ \int \frac{1}{{\left (c x^{2} + a\right )} \sqrt{e x + d}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*x^2 + a)*sqrt(e*x + d)), x)

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Fricas [A]  time = 0.227353, size = 1270, normalized size = 2.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right ) \sqrt{d + e x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + c*x**2)*sqrt(d + e*x)), x)

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GIAC/XCAS [A]  time = 8.98433, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done