3.528 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=153 \[ \frac{3 \sqrt{c} \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^4}+\frac{3 c d \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{3 c \sqrt{a+c x^2} (2 d-e x)}{2 e^3}-\frac{\left (a+c x^2\right )^{3/2}}{e (d+e x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.390144, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{3 \sqrt{c} \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 e^4}+\frac{3 c d \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^4}-\frac{3 c \sqrt{a+c x^2} (2 d-e x)}{2 e^3}-\frac{\left (a+c x^2\right )^{3/2}}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.1389, size = 139, normalized size = 0.91 \[ \frac{3 \sqrt{c} \left (a e^{2} + 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{2 e^{4}} + \frac{3 c d \sqrt{a e^{2} + c d^{2}} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{e^{4}} - \frac{3 c \sqrt{a + c x^{2}} \left (2 d - e x\right )}{2 e^{3}} - \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.297749, size = 179, normalized size = 1.17 \[ \frac{-\frac{e \sqrt{a+c x^2} \left (2 a e^2+c \left (6 d^2+3 d e x-e^2 x^2\right )\right )}{d+e x}+3 \sqrt{c} \left (a e^2+2 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+6 c d \sqrt{a e^2+c d^2} \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )-6 c d \sqrt{a e^2+c d^2} \log (d+e x)}{2 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^2,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.016, size = 1154, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^2,x)

[Out]

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

_______________________________________________________________________________________

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((c*x^2 + a)^(3/2)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.586023, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^2,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**2,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**2, x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^2,x, algorithm="giac")

[Out]

undef