3.530 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=200 \[ \frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac{c \sqrt{a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 0.515471, antiderivative size = 200, 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 \[ \frac{c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}+\frac{c^2 d \left (3 a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \left (a e^2+c d^2\right )^{3/2}}-\frac{c \sqrt{a+c x^2} \left (e x \left (2 a e^2+3 c d^2\right )+d \left (a e^2+2 c d^2\right )\right )}{2 e^3 (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{\left (a+c x^2\right )^{3/2}}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 49.7698, size = 182, normalized size = 0.91 \[ \frac{c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{4}} + \frac{c^{2} d \left (3 a e^{2} + 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 e^{4} \left (a e^{2} + c d^{2}\right )^{\frac{3}{2}}} - \frac{c \sqrt{a + c x^{2}} \left (d \left (a e^{2} + 2 c d^{2}\right ) + e x \left (2 a e^{2} + 3 c d^{2}\right )\right )}{2 e^{3} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} - \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.467897, size = 242, normalized size = 1.21 \[ \frac{-\frac{e \sqrt{a+c x^2} \left (2 a^2 e^4+a c e^2 \left (5 d^2+9 d e x+8 e^2 x^2\right )+c^2 d^2 \left (6 d^2+15 d e x+11 e^2 x^2\right )\right )}{(d+e x)^3 \left (a e^2+c d^2\right )}+6 c^{3/2} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\frac{3 c^2 d \left (3 a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{3/2}}-\frac{3 c^2 d \left (3 a e^2+2 c d^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{3/2}}}{6 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^4,x]

[Out]

(-((e*Sqrt[a + c*x^2]*(2*a^2*e^4 + a*c*e^2*(5*d^2 + 9*d*e*x + 8*e^2*x^2) + c^2*d
^2*(6*d^2 + 15*d*e*x + 11*e^2*x^2)))/((c*d^2 + a*e^2)*(d + e*x)^3)) - (3*c^2*d*(
2*c*d^2 + 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(3/2) + 6*c^(3/2)*Log[c*x + Sqr
t[c]*Sqrt[a + c*x^2]] + (3*c^2*d*(2*c*d^2 + 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^
2 + a*e^2]*Sqrt[a + c*x^2]])/(c*d^2 + a*e^2)^(3/2))/(6*e^4)

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Maple [B]  time = 0.021, size = 2490, normalized size = 12.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^4,x)

[Out]

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

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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((c*x^2 + a)^(3/2)/(e*x + d)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 7.14981, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^4,x, algorithm="fricas")

[Out]

[1/12*(6*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^2*e^3 + a*c*e^5)*x^3 + 3*(c^2*d^3*e^2 +
a*c*d*e^4)*x^2 + 3*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c)*log
(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - 2*(6*c^2*d^4*e + 5*a*c*d^2*e^3 +
2*a^2*e^5 + (11*c^2*d^2*e^3 + 8*a*c*e^5)*x^2 + 3*(5*c^2*d^3*e^2 + 3*a*c*d*e^4)*x
)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + (2*c^3*
d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(2*c^
3*d^5*e + 3*a*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*d^5*e^4 + a*d^3*e^6
+ (c*d^2*e^7 + a*e^9)*x^3 + 3*(c*d^3*e^6 + a*d*e^8)*x^2 + 3*(c*d^4*e^5 + a*d^2*e
^7)*x)*sqrt(c*d^2 + a*e^2)), 1/6*(3*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^2*e^3 + a*c*
e^5)*x^3 + 3*(c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(c^2*d^4*e + a*c*d^2*e^3)*x)*sqrt
(-c*d^2 - a*e^2)*sqrt(c)*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) - (6*c^
2*d^4*e + 5*a*c*d^2*e^3 + 2*a^2*e^5 + (11*c^2*d^2*e^3 + 8*a*c*e^5)*x^2 + 3*(5*c^
2*d^3*e^2 + 3*a*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 3*(2*c^3*d^6
+ 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 + 3*a
*c^2*d^2*e^4)*x^2 + 3*(2*c^3*d^5*e + 3*a*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*d^5*e^4 + a*d^3*e^6 +
(c*d^2*e^7 + a*e^9)*x^3 + 3*(c*d^3*e^6 + a*d*e^8)*x^2 + 3*(c*d^4*e^5 + a*d^2*e^
7)*x)*sqrt(-c*d^2 - a*e^2)), 1/12*(12*(c^2*d^5 + a*c*d^3*e^2 + (c^2*d^2*e^3 + a*
c*e^5)*x^3 + 3*(c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(c^2*d^4*e + a*c*d^2*e^3)*x)*sq
rt(c*d^2 + a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(-c))) - 2*(6*c^2*d^4
*e + 5*a*c*d^2*e^3 + 2*a^2*e^5 + (11*c^2*d^2*e^3 + 8*a*c*e^5)*x^2 + 3*(5*c^2*d^3
*e^2 + 3*a*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*(2*c^3*d^6 + 3*a*
c^2*d^4*e^2 + (2*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*d^4*e^2 + 3*a*c^2*d
^2*e^4)*x^2 + 3*(2*c^3*d^5*e + 3*a*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
*d^5*e^4 + a*d^3*e^6 + (c*d^2*e^7 + a*e^9)*x^3 + 3*(c*d^3*e^6 + a*d*e^8)*x^2 + 3
*(c*d^4*e^5 + a*d^2*e^7)*x)*sqrt(c*d^2 + a*e^2)), 1/6*(6*(c^2*d^5 + a*c*d^3*e^2
+ (c^2*d^2*e^3 + a*c*e^5)*x^3 + 3*(c^2*d^3*e^2 + a*c*d*e^4)*x^2 + 3*(c^2*d^4*e +
a*c*d^2*e^3)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2 + a)*sqrt(
-c))) - (6*c^2*d^4*e + 5*a*c*d^2*e^3 + 2*a^2*e^5 + (11*c^2*d^2*e^3 + 8*a*c*e^5)*
x^2 + 3*(5*c^2*d^3*e^2 + 3*a*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) -
3*(2*c^3*d^6 + 3*a*c^2*d^4*e^2 + (2*c^3*d^3*e^3 + 3*a*c^2*d*e^5)*x^3 + 3*(2*c^3*
d^4*e^2 + 3*a*c^2*d^2*e^4)*x^2 + 3*(2*c^3*d^5*e + 3*a*c^2*d^3*e^3)*x)*arctan(sqr
t(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((c*d^5*e^4
+ a*d^3*e^6 + (c*d^2*e^7 + a*e^9)*x^3 + 3*(c*d^3*e^6 + a*d*e^8)*x^2 + 3*(c*d^4*e
^5 + a*d^2*e^7)*x)*sqrt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**4,x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**4, x)

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GIAC/XCAS [A]  time = 0.61567, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^4,x, algorithm="giac")

[Out]

sage0*x