3.522 \(\int \frac{\sqrt{a+c x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=206 \[ -\frac{a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

[Out]

-(c*(4*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e
*x)^2) - (e*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (5*c*d*e*(a + c
*x^2)^(3/2))/(12*(c*d^2 + a*e^2)^2*(d + e*x)^3) - (a*c^2*(4*c*d^2 - a*e^2)*ArcTa
nh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2
))

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Rubi [A]  time = 0.385787, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{a c^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{8 \left (a e^2+c d^2\right )^{7/2}}-\frac{5 c d e \left (a+c x^2\right )^{3/2}}{12 (d+e x)^3 \left (a e^2+c d^2\right )^2}-\frac{c \sqrt{a+c x^2} \left (4 c d^2-a e^2\right ) (a e-c d x)}{8 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{3/2}}{4 (d+e x)^4 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^2]/(d + e*x)^5,x]

[Out]

-(c*(4*c*d^2 - a*e^2)*(a*e - c*d*x)*Sqrt[a + c*x^2])/(8*(c*d^2 + a*e^2)^3*(d + e
*x)^2) - (e*(a + c*x^2)^(3/2))/(4*(c*d^2 + a*e^2)*(d + e*x)^4) - (5*c*d*e*(a + c
*x^2)^(3/2))/(12*(c*d^2 + a*e^2)^2*(d + e*x)^3) - (a*c^2*(4*c*d^2 - a*e^2)*ArcTa
nh[(a*e - c*d*x)/(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2])])/(8*(c*d^2 + a*e^2)^(7/2
))

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Rubi in Sympy [A]  time = 36.7381, size = 189, normalized size = 0.92 \[ \frac{a c^{2} \left (a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{8 \left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} - \frac{5 c d e \left (a + c x^{2}\right )^{\frac{3}{2}}}{12 \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{c \sqrt{a + c x^{2}} \left (2 a e - 2 c d x\right ) \left (a e^{2} - 4 c d^{2}\right )}{16 \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 \left (d + e x\right )^{4} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(1/2)/(e*x+d)**5,x)

[Out]

a*c**2*(a*e**2 - 4*c*d**2)*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c
*d**2)))/(8*(a*e**2 + c*d**2)**(7/2)) - 5*c*d*e*(a + c*x**2)**(3/2)/(12*(d + e*x
)**3*(a*e**2 + c*d**2)**2) + c*sqrt(a + c*x**2)*(2*a*e - 2*c*d*x)*(a*e**2 - 4*c*
d**2)/(16*(d + e*x)**2*(a*e**2 + c*d**2)**3) - e*(a + c*x**2)**(3/2)/(4*(d + e*x
)**4*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.390948, size = 248, normalized size = 1.2 \[ \frac{\sqrt{a+c x^2} \sqrt{a e^2+c d^2} \left (c^2 d (d+e x)^3 \left (2 c d^2-13 a e^2\right )+2 c d (d+e x) \left (a e^2+c d^2\right )^2+c (d+e x)^2 \left (2 c d^2-3 a e^2\right ) \left (a e^2+c d^2\right )-6 \left (a e^2+c d^2\right )^3\right )+3 a c^2 e (d+e x)^4 \left (a e^2-4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-3 a c^2 e (d+e x)^4 \left (a e^2-4 c d^2\right ) \log (d+e x)}{24 e (d+e x)^4 \left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^2]/(d + e*x)^5,x]

[Out]

(Sqrt[c*d^2 + a*e^2]*Sqrt[a + c*x^2]*(-6*(c*d^2 + a*e^2)^3 + 2*c*d*(c*d^2 + a*e^
2)^2*(d + e*x) + c*(2*c*d^2 - 3*a*e^2)*(c*d^2 + a*e^2)*(d + e*x)^2 + c^2*d*(2*c*
d^2 - 13*a*e^2)*(d + e*x)^3) - 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[d +
e*x] + 3*a*c^2*e*(-4*c*d^2 + a*e^2)*(d + e*x)^4*Log[a*e - c*d*x + Sqrt[c*d^2 + a
*e^2]*Sqrt[a + c*x^2]])/(24*e*(c*d^2 + a*e^2)^(7/2)*(d + e*x)^4)

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Maple [B]  time = 0.023, size = 2073, normalized size = 10.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(1/2)/(e*x+d)^5,x)

[Out]

-5/12/e^2*c*d/(a*e^2+c*d^2)^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)-5/8/e*c^2*d^2/(a*e^2+c*d^2)^3/(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)-5/8*c^3*d^3/(a*e^2+c*d^2)^4/(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)+5/8/e^2*c^(9/2)*d^5/(a*e^2+c*d^2)^4*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
/8/(a*e^2+c*d^2)^3*c^2*d/(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)-1/8/(a*e^2+c*d^2)^3*c^3*d*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)
^(1/2)*x+1/8/e^2/(a*e^2+c*d^2)^2*c^(5/2)*d*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))+1/8/e/(a*e^2+c*d^2)^2*c^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+1/8/e^
3/(a*e^2+c*d^2)^2*c^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))*d^2+5/8*c^(7/2)*d^3/(a*e^2+c*d^2)^4*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-3/4/e*c^3*d^2/(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-1/8/(a*e^2+c*d^2)^3*c^(5/2)*d*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-1/8/e/(a*e^2+c*d^2)^2*c^2*(c*(
d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)+1/8/e/(a*e^2+c*d^2)^2*c/(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)+5/8/e^3*c^5*d^6/(a*e^2
+c*d^2)^4/((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))+5/8*c^4*d^3/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e
^2)^(1/2)*x-3/4/e^2*c^(7/2)*d^3/(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))-3/4/e^3*c^4*d^4/(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
))-1/4/e^3/(a*e^2+c*d^2)/(d/e+x)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^
2)^(3/2)+3/4/e*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)-5/8/e*c^4*d^4/(a*e^2+c*d^2)^4*(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c
*d^2)/e^2)^(1/2)+5/8/e*c^4*d^4/(a*e^2+c*d^2)^4/((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

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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)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.06111, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/(e*x + d)^5,x, algorithm="fricas")

[Out]

[-1/48*(2*(28*a*c^2*d^4*e + 19*a^2*c*d^2*e^3 + 6*a^3*e^5 - (2*c^3*d^3*e^2 - 13*a
*c^2*d*e^4)*x^3 - (8*c^3*d^4*e - 40*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 - (12*c^3*d
^5 - 37*a*c^2*d^3*e^2 - 4*a^2*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) +
3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 4*(4*a*
c^3*d^3*e^3 - a^2*c^2*d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a^2*c^2*d^2*e^4)*x^2 + 4
*(4*a*c^3*d^5*e - a^2*c^2*d^3*e^3)*x)*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^3*d^10 + 3*a
*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4 + a^3*d^4*e^6 + (c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 +
3*a^2*c*d^2*e^8 + a^3*e^10)*x^4 + 4*(c^3*d^7*e^3 + 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3
*e^7 + a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 + a^3
*d^2*e^8)*x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*
x)*sqrt(c*d^2 + a*e^2)), -1/24*((28*a*c^2*d^4*e + 19*a^2*c*d^2*e^3 + 6*a^3*e^5 -
(2*c^3*d^3*e^2 - 13*a*c^2*d*e^4)*x^3 - (8*c^3*d^4*e - 40*a*c^2*d^2*e^3 - 3*a^2*
c*e^5)*x^2 - (12*c^3*d^5 - 37*a*c^2*d^3*e^2 - 4*a^2*c*d*e^4)*x)*sqrt(-c*d^2 - a*
e^2)*sqrt(c*x^2 + a) - 3*(4*a*c^3*d^6 - a^2*c^2*d^4*e^2 + (4*a*c^3*d^2*e^4 - a^2
*c^2*e^6)*x^4 + 4*(4*a*c^3*d^3*e^3 - a^2*c^2*d*e^5)*x^3 + 6*(4*a*c^3*d^4*e^2 - a
^2*c^2*d^2*e^4)*x^2 + 4*(4*a*c^3*d^5*e - a^2*c^2*d^3*e^3)*x)*arctan(sqrt(-c*d^2
- a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c^3*d^10 + 3*a*c^2*
d^8*e^2 + 3*a^2*c*d^6*e^4 + a^3*d^4*e^6 + (c^3*d^6*e^4 + 3*a*c^2*d^4*e^6 + 3*a^2
*c*d^2*e^8 + a^3*e^10)*x^4 + 4*(c^3*d^7*e^3 + 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7
+ a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 + 3*a*c^2*d^6*e^4 + 3*a^2*c*d^4*e^6 + a^3*d^2*
e^8)*x^2 + 4*(c^3*d^9*e + 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 + a^3*d^3*e^7)*x)*sq
rt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + c x^{2}}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt(a + c*x**2)/(d + e*x)**5, x)

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GIAC/XCAS [A]  time = 1.23342, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + a)/(e*x + d)^5,x, algorithm="giac")

[Out]

Done