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

Optimal. Leaf size=675 \[ \frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )} \]

[Out]

(x*Sqrt[d + e*x])/(2*a*(a + c*x^2)) + (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTa
nh[(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]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt
[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTan
h[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[
Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c])*Log[Sqr
t[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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[
c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

Rubi [A]  time = 2.58735, antiderivative size = 675, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (\sqrt{a e^2+c d^2}+\sqrt{c} d\right ) \tanh ^{-1}\left (\frac{\sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{2} \sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}\right )}{4 \sqrt{2} a c^{3/4} \sqrt{a e^2+c d^2} \sqrt{\sqrt{c} d-\sqrt{a e^2+c d^2}}}-\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (-\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{e \left (d-\frac{\sqrt{a e^2+c d^2}}{\sqrt{c}}\right ) \log \left (\sqrt{2} \sqrt [4]{c} \sqrt{d+e x} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}+\sqrt{a e^2+c d^2}+\sqrt{c} (d+e x)\right )}{8 \sqrt{2} a \sqrt [4]{c} \sqrt{a e^2+c d^2} \sqrt{\sqrt{a e^2+c d^2}+\sqrt{c} d}}+\frac{x \sqrt{d+e x}}{2 a \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[d + e*x])/(2*a*(a + c*x^2)) + (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTa
nh[(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]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt
[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(Sqrt[c]*d + Sqrt[c*d^2 + a*e^2])*ArcTan
h[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(3/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[
Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c])*Log[Sqr
t[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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[
c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(d - Sqrt[c*d^2 + a*e^2]/Sqrt[c])*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)])/(8*Sqrt[2]*a*c^(1/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d
+ Sqrt[c*d^2 + a*e^2]])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 0.478957, size = 208, normalized size = 0.31 \[ \frac{\frac{2 \sqrt{a} x \sqrt{d+e x}}{a+c x^2}-\frac{\left (\sqrt{a} \sqrt{c} e+2 i c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-i \sqrt{a} \sqrt{c} e}}\right )}{c \sqrt{c d-i \sqrt{a} \sqrt{c} e}}-\frac{\left (\sqrt{a} \sqrt{c} e-2 i c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d+i \sqrt{a} \sqrt{c} e}}\right )}{c \sqrt{c d+i \sqrt{a} \sqrt{c} e}}}{4 a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*x*Sqrt[d + e*x])/(a + c*x^2) - (((2*I)*c*d + Sqrt[a]*Sqrt[c]*e)*ArcT
anh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - I*Sqrt[a]*Sqrt[c]*e]])/(c*Sqrt[c*d - I*Sq
rt[a]*Sqrt[c]*e]) - (((-2*I)*c*d + Sqrt[a]*Sqrt[c]*e)*ArcTanh[(Sqrt[c]*Sqrt[d +
e*x])/Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]])/(c*Sqrt[c*d + I*Sqrt[a]*Sqrt[c]*e]))/(4*
a^(3/2))

_______________________________________________________________________________________

Maple [F]  time = 1.119, size = 0, normalized size = 0. \[ \int{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}}\sqrt{ex+d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.23676, size = 1851, normalized size = 2.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((a*c*x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt
(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2
))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 - (2*a^3*c^4*d^4 + 3*a
^4*c^3*d^2*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c
^3*e^4)))*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*
c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*
x^2 + a^2)*sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3
*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*
c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (a^2*c*d*e^4 - (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2
*e^2 + a^5*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*
sqrt(-(4*c*d^3 + 3*a*d*e^2 + (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 +
2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) + (a*c*x^2 + a^2)
*sqrt(-(4*c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 +
2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3
+ a*e^5)*sqrt(e*x + d) + (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5
*c^2*e^4)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c
*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*
d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) - (a*c*x^2 + a^2)*sqrt(-(4*
c*d^3 + 3*a*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4
*d^2*e^2 + a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))*log((4*c*d^2*e^3 + a*e^5)*s
qrt(e*x + d) - (a^2*c*d*e^4 + (2*a^3*c^4*d^4 + 3*a^4*c^3*d^2*e^2 + a^5*c^2*e^4)*
sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 + a^5*c^3*e^4)))*sqrt(-(4*c*d^3 + 3*a
*d*e^2 - (a^3*c^2*d^2 + a^4*c*e^2)*sqrt(-e^6/(a^3*c^5*d^4 + 2*a^4*c^4*d^2*e^2 +
a^5*c^3*e^4)))/(a^3*c^2*d^2 + a^4*c*e^2))) + 4*sqrt(e*x + d)*x)/(a*c*x^2 + a^2)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError