3.521 \(\int \frac{\sqrt{a+c x^2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=144 \[ -\frac{a c^2 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}}-\frac{c d \sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]

[Out]

-(c*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(2*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (e*(a +
c*x^2)^(3/2))/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (a*c^2*d*ArcTanh[(a*e - c*d*x)/
(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(5/2))

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Rubi [A]  time = 0.22659, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{a c^2 d \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{5/2}}-\frac{c d \sqrt{a+c x^2} (a e-c d x)}{2 (d+e x)^2 \left (a e^2+c d^2\right )^2}-\frac{e \left (a+c x^2\right )^{3/2}}{3 (d+e x)^3 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(c*d*(a*e - c*d*x)*Sqrt[a + c*x^2])/(2*(c*d^2 + a*e^2)^2*(d + e*x)^2) - (e*(a +
c*x^2)^(3/2))/(3*(c*d^2 + a*e^2)*(d + e*x)^3) - (a*c^2*d*ArcTanh[(a*e - c*d*x)/
(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(2*(c*d^2 + a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 22.6004, size = 133, normalized size = 0.92 \[ - \frac{a c^{2} d \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{5}{2}}} - \frac{c d \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right )}{4 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}}}{3 \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(1/2)/(e*x+d)**4,x)

[Out]

-a*c**2*d*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(2*(a*e*
*2 + c*d**2)**(5/2)) - c*d*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)/(4*(d + e*x)**2*(a
*e**2 + c*d**2)**2) - e*(a + c*x**2)**(3/2)/(3*(d + e*x)**3*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.268935, size = 173, normalized size = 1.2 \[ \frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (-2 a^2 e^3-a c e \left (5 d^2+3 d e x+2 e^2 x^2\right )+c^2 d^2 x (3 d+e x)\right )-3 a c^2 d (d+e x)^3 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )+3 a c^2 d (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left (a e^2+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^2]/(d + e*x)^4,x]

[Out]

(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-2*a^2*e^3 + c^2*d^2*x*(3*d + e*x) - a*c*e
*(5*d^2 + 3*d*e*x + 2*e^2*x^2)) + 3*a*c^2*d*(d + e*x)^3*Log[d + e*x] - 3*a*c^2*d
*(d + e*x)^3*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]])/(6*(c*d^2 +
a*e^2)^(5/2)*(d + e*x)^3)

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Maple [B]  time = 0.018, size = 1262, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(1/2)/(e*x+d)^4,x)

[Out]

-1/3/e^2/(a*e^2+c*d^2)/(d/e+x)^3*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(3/2)-1/2/e*c*d/(a*e^2+c*d^2)^2/(d/e+x)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(3/2)-1/2*c^2*d^2/(a*e^2+c*d^2)^3/(d/e+x)*(c*(d/e+x)^2-2*c*d/e*(d/e+x
)+(a*e^2+c*d^2)/e^2)^(3/2)-1/2/e*c^3*d^3/(a*e^2+c*d^2)^3*(c*(d/e+x)^2-2*c*d/e*(d
/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+1/2/e^2*c^(7/2)*d^4/(a*e^2+c*d^2)^3*ln((-c*d/e+c*
(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))+1/2/e*c^
3*d^3/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*
(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(1/2))/(d/e+x))*a+1/2/e^3*c^4*d^5/(a*e^2+c*d^2)^3/((a*e^2+c*d^2)/e^2)^(1/2)*
ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))+1/2*c^3*d^2/(a*e^2+c*d^2)^3*
(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x+1/2*c^(5/2)*d^2/(a*e^2+c
*d^2)^3*ln((-c*d/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(1/2))*a+1/2/e*c^2*d/(a*e^2+c*d^2)^2*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(1/2)-1/2/e^2*c^(5/2)*d^2/(a*e^2+c*d^2)^2*ln((-c*d/e+c*(d/e+x))/c^(1/
2)+(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))-1/2/e*c^2*d/(a*e^2+c*d
^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^
2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+
x))*a-1/2/e^3*c^3*d^3/(a*e^2+c*d^2)^2/((a*e^2+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2+c*d
^2)/e^2-2*c*d/e*(d/e+x)+2*((a*e^2+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2-2*c*d/e*(d/e+x)
+(a*e^2+c*d^2)/e^2)^(1/2))/(d/e+x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.527966, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (5 \, a c d^{2} e + 2 \, a^{2} e^{3} -{\left (c^{2} d^{2} e - 2 \, a c e^{3}\right )} x^{2} - 3 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{c d^{2} + a e^{2}} \sqrt{c x^{2} + a} - 3 \,{\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + a c^{2} d^{4}\right )} \log \left (\frac{{\left (2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \sqrt{c d^{2} + a e^{2}} + 2 \,{\left (a c d^{2} e + a^{2} e^{3} -{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{12 \,{\left (c^{2} d^{7} + 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} +{\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} + 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{c d^{2} + a e^{2}}}, -\frac{{\left (5 \, a c d^{2} e + 2 \, a^{2} e^{3} -{\left (c^{2} d^{2} e - 2 \, a c e^{3}\right )} x^{2} - 3 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}} \sqrt{c x^{2} + a} - 3 \,{\left (a c^{2} d e^{3} x^{3} + 3 \, a c^{2} d^{2} e^{2} x^{2} + 3 \, a c^{2} d^{3} e x + a c^{2} d^{4}\right )} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{c x^{2} + a}}\right )}{6 \,{\left (c^{2} d^{7} + 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} +{\left (c^{2} d^{4} e^{3} + 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} + 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e + 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{-c d^{2} - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/(e*x + d)^4,x, algorithm="fricas")

[Out]

[-1/12*(2*(5*a*c*d^2*e + 2*a^2*e^3 - (c^2*d^2*e - 2*a*c*e^3)*x^2 - 3*(c^2*d^3 -
a*c*d*e^2)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) - 3*(a*c^2*d*e^3*x^3 + 3*a*c^2
*d^2*e^2*x^2 + 3*a*c^2*d^3*e*x + a*c^2*d^4)*log(((2*a*c*d*e*x - a*c*d^2 - 2*a^2*
e^2 - (2*c^2*d^2 + a*c*e^2)*x^2)*sqrt(c*d^2 + a*e^2) + 2*(a*c*d^2*e + a^2*e^3 -
(c^2*d^3 + a*c*d*e^2)*x)*sqrt(c*x^2 + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^7
+ 2*a*c*d^5*e^2 + a^2*d^3*e^4 + (c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*
(c^2*d^5*e^2 + 2*a*c*d^3*e^4 + a^2*d*e^6)*x^2 + 3*(c^2*d^6*e + 2*a*c*d^4*e^3 + a
^2*d^2*e^5)*x)*sqrt(c*d^2 + a*e^2)), -1/6*((5*a*c*d^2*e + 2*a^2*e^3 - (c^2*d^2*e
- 2*a*c*e^3)*x^2 - 3*(c^2*d^3 - a*c*d*e^2)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 +
a) - 3*(a*c^2*d*e^3*x^3 + 3*a*c^2*d^2*e^2*x^2 + 3*a*c^2*d^3*e*x + a*c^2*d^4)*ar
ctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c^
2*d^7 + 2*a*c*d^5*e^2 + a^2*d^3*e^4 + (c^2*d^4*e^3 + 2*a*c*d^2*e^5 + a^2*e^7)*x^
3 + 3*(c^2*d^5*e^2 + 2*a*c*d^3*e^4 + a^2*d*e^6)*x^2 + 3*(c^2*d^6*e + 2*a*c*d^4*e
^3 + a^2*d^2*e^5)*x)*sqrt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(1/2)/(e*x+d)**4,x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.597084, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/(e*x + d)^4,x, algorithm="giac")

[Out]

sage0*x