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

Optimal. Leaf size=161 \[ -\frac{3 c^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{3 c \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \sqrt{a e^2+c d^2}}+\frac{3 c \sqrt{a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 0.37997, antiderivative size = 161, 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 \[ -\frac{3 c^{3/2} d \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{e^4}-\frac{3 c \left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 e^4 \sqrt{a e^2+c d^2}}+\frac{3 c \sqrt{a+c x^2} (2 d+e x)}{2 e^3 (d+e x)}-\frac{\left (a+c x^2\right )^{3/2}}{2 e (d+e x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.3061, size = 148, normalized size = 0.92 \[ - \frac{3 c^{\frac{3}{2}} d \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{e^{4}} + \frac{3 c \sqrt{a + c x^{2}} \left (2 d + e x\right )}{2 e^{3} \left (d + e x\right )} - \frac{3 c \left (a e^{2} + 2 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 e^{4} \sqrt{a e^{2} + c d^{2}}} - \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.367924, size = 189, normalized size = 1.17 \[ \frac{-6 c^{3/2} d \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\frac{e \sqrt{a+c x^2} \left (c \left (6 d^2+9 d e x+2 e^2 x^2\right )-a e^2\right )}{(d+e x)^2}-\frac{3 c \left (a e^2+2 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\sqrt{a e^2+c d^2}}+\frac{3 c \left (a e^2+2 c d^2\right ) \log (d+e x)}{\sqrt{a e^2+c d^2}}}{2 e^4} \]

Antiderivative was successfully verified.

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

[Out]

((e*Sqrt[a + c*x^2]*(-(a*e^2) + c*(6*d^2 + 9*d*e*x + 2*e^2*x^2)))/(d + e*x)^2 +
(3*c*(2*c*d^2 + a*e^2)*Log[d + e*x])/Sqrt[c*d^2 + a*e^2] - 6*c^(3/2)*d*Log[c*x +
Sqrt[c]*Sqrt[a + c*x^2]] - (3*c*(2*c*d^2 + a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2
+ a*e^2]*Sqrt[a + c*x^2]])/Sqrt[c*d^2 + a*e^2])/(2*e^4)

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Maple [B]  time = 0.017, size = 2117, normalized size = 13.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.720789, 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)^3,x, algorithm="fricas")

[Out]

[1/4*(6*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*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*(2*c*e^3*x^2 + 9*c*d*e^2*x + 6*c*d^2
*e - a*e^3)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*(2*c^2*d^4 + a*c*d^2*e^2 + (
2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*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)))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(c*d^2 + a*e^2)), -1/4*(12*(c*d*e^2
*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(c*d^2 + a*e^2)*sqrt(-c)*arctan(c*x/(sqrt(c*x^2
+ a)*sqrt(-c))) - 2*(2*c*e^3*x^2 + 9*c*d*e^2*x + 6*c*d^2*e - a*e^3)*sqrt(c*d^2 +
a*e^2)*sqrt(c*x^2 + a) - 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)
*x^2 + 2*(2*c^2*d^3*e + a*c*d*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)))/((e^6*x^2 + 2*d*
e^5*x + d^2*e^4)*sqrt(c*d^2 + a*e^2)), 1/2*(3*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3
)*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*c*e^3*x^2 + 9*c*d*e^2*x + 6*c*d^2*e - a*e^3)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2
+ a) + 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2 + a*c*e^4)*x^2 + 2*(2*c^2*d^
3*e + a*c*d*e^3)*x)*arctan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*s
qrt(c*x^2 + a))))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*sqrt(-c*d^2 - a*e^2)), -1/2*(
6*(c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*sqrt(-c*d^2 - a*e^2)*sqrt(-c)*arctan(c*x/(
sqrt(c*x^2 + a)*sqrt(-c))) - (2*c*e^3*x^2 + 9*c*d*e^2*x + 6*c*d^2*e - a*e^3)*sqr
t(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) - 3*(2*c^2*d^4 + a*c*d^2*e^2 + (2*c^2*d^2*e^2
+ a*c*e^4)*x^2 + 2*(2*c^2*d^3*e + a*c*d*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))))/((e^6*x^2 + 2*d*e^5*x + d^2*e^4)*s
qrt(-c*d^2 - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.577435, 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)^3,x, algorithm="giac")

[Out]

sage0*x