3.565 \(\int \frac{1}{(d+e x)^4 \left (a+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=293 \[ \frac{c e \sqrt{a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac{5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac{c d e \sqrt{a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )} \]

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^3*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 4*
a*e^2)*Sqrt[a + c*x^2])/(3*a*(c*d^2 + a*e^2)^2*(d + e*x)^3) + (c*d*e*(6*c*d^2 -
29*a*e^2)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*e*(6*c^2*d^4
- 83*a*c*d^2*e^2 + 16*a^2*e^4)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^4*(d + e*x
)) - (5*c^2*d*e^2*(4*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*(c*d^2 + a*e^2)^(9/2))

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Rubi [A]  time = 1.06755, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{c e \sqrt{a+c x^2} \left (16 a^2 e^4-83 a c d^2 e^2+6 c^2 d^4\right )}{6 a (d+e x) \left (a e^2+c d^2\right )^4}-\frac{5 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}}+\frac{c d e \sqrt{a+c x^2} \left (6 c d^2-29 a e^2\right )}{6 a (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (3 c d^2-4 a e^2\right )}{3 a (d+e x)^3 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^3 \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^4*(a + c*x^2)^(3/2)),x]

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*(d + e*x)^3*Sqrt[a + c*x^2]) + (e*(3*c*d^2 - 4*
a*e^2)*Sqrt[a + c*x^2])/(3*a*(c*d^2 + a*e^2)^2*(d + e*x)^3) + (c*d*e*(6*c*d^2 -
29*a*e^2)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*e*(6*c^2*d^4
- 83*a*c*d^2*e^2 + 16*a^2*e^4)*Sqrt[a + c*x^2])/(6*a*(c*d^2 + a*e^2)^4*(d + e*x
)) - (5*c^2*d*e^2*(4*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*(c*d^2 + a*e^2)^(9/2))

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Rubi in Sympy [A]  time = 109.533, size = 270, normalized size = 0.92 \[ \frac{5 c^{2} d e^{2} \left (3 a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{9}{2}}} - \frac{c d e \sqrt{a + c x^{2}} \left (29 a e^{2} - 6 c d^{2}\right )}{6 a \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{3}} + \frac{c e \sqrt{a + c x^{2}} \left (16 a^{2} e^{4} - 83 a c d^{2} e^{2} + 6 c^{2} d^{4}\right )}{6 a \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{4}} - \frac{e \sqrt{a + c x^{2}} \left (4 a e^{2} - 3 c d^{2}\right )}{3 a \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{a \sqrt{a + c x^{2}} \left (d + e x\right )^{3} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**4/(c*x**2+a)**(3/2),x)

[Out]

5*c**2*d*e**2*(3*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)))/(2*(a*e**2 + c*d**2)**(9/2)) - c*d*e*sqrt(a + c*x**2)*(29*a*e*
*2 - 6*c*d**2)/(6*a*(d + e*x)**2*(a*e**2 + c*d**2)**3) + c*e*sqrt(a + c*x**2)*(1
6*a**2*e**4 - 83*a*c*d**2*e**2 + 6*c**2*d**4)/(6*a*(d + e*x)*(a*e**2 + c*d**2)**
4) - e*sqrt(a + c*x**2)*(4*a*e**2 - 3*c*d**2)/(3*a*(d + e*x)**3*(a*e**2 + c*d**2
)**2) + (a*e + c*d*x)/(a*sqrt(a + c*x**2)*(d + e*x)**3*(a*e**2 + c*d**2))

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Mathematica [A]  time = 1.45505, size = 279, normalized size = 0.95 \[ \frac{1}{6} \left (\frac{\sqrt{a+c x^2} \left (\frac{6 c^2 \left (a^2 e^3 (e x-4 d)+2 a c d^2 e (2 d-3 e x)+c^2 d^4 x\right )}{a \left (a+c x^2\right )}+\frac{c e^3 \left (10 a e^2-47 c d^2\right )}{d+e x}-\frac{11 c d e^3 \left (a e^2+c d^2\right )}{(d+e x)^2}-\frac{2 e^3 \left (a e^2+c d^2\right )^2}{(d+e x)^3}\right )}{\left (a e^2+c d^2\right )^4}-\frac{15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 c^2 d e^2 \left (4 c d^2-3 a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^4*(a + c*x^2)^(3/2)),x]

[Out]

((Sqrt[a + c*x^2]*((-2*e^3*(c*d^2 + a*e^2)^2)/(d + e*x)^3 - (11*c*d*e^3*(c*d^2 +
a*e^2))/(d + e*x)^2 + (c*e^3*(-47*c*d^2 + 10*a*e^2))/(d + e*x) + (6*c^2*(c^2*d^
4*x + 2*a*c*d^2*e*(2*d - 3*e*x) + a^2*e^3*(-4*d + e*x)))/(a*(a + c*x^2))))/(c*d^
2 + a*e^2)^4 + (15*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[d + e*x])/(c*d^2 + a*e^2)^(
9/2) - (15*c^2*d*e^2*(4*c*d^2 - 3*a*e^2)*Log[a*e - c*d*x + Sqrt[c*d^2 + a*e^2]*S
qrt[a + c*x^2]])/(c*d^2 + a*e^2)^(9/2))/6

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Maple [B]  time = 0.024, size = 898, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^4/(c*x^2+a)^(3/2),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)
^(1/2)-7/6/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)^(1/2)-35/6*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)^(1/2)+35/2*e*c^3*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)+35/2*c^4*d^4/(a*e^2+c*d^2)^4/a/(c*(d/e+x)^2-2*c
*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(1/2)*x-35/2*e*c^3*d^3/(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))-115/6*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-15/
2*e*c^2*d/(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)+
15/2*e*c^2*d/(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))+4/3/(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)^(1/2)+8/3/(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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.30659, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)^4),x, algorithm="fricas")

[Out]

[1/12*(2*(24*a*c^3*d^6*e - 84*a^2*c^2*d^4*e^3 - 5*a^3*c*d^2*e^5 - 2*a^4*e^7 + (6
*c^4*d^4*e^3 - 83*a*c^3*d^2*e^5 + 16*a^2*c^2*e^7)*x^4 + 3*(6*c^4*d^5*e^2 - 63*a*
c^3*d^3*e^4 + a^2*c^2*d*e^6)*x^3 + 2*(9*c^4*d^6*e - 48*a*c^3*d^4*e^3 - 53*a^2*c^
2*d^2*e^5 + 4*a^3*c*e^7)*x^2 + 3*(2*c^4*d^7 + 12*a*c^3*d^5*e^2 - 57*a^2*c^2*d^3*
e^4 + 3*a^3*c*d*e^6)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 15*(4*a^2*c^3*d^6*
e^2 - 3*a^3*c^2*d^4*e^4 + (4*a*c^4*d^3*e^5 - 3*a^2*c^3*d*e^7)*x^5 + 3*(4*a*c^4*d
^4*e^4 - 3*a^2*c^3*d^2*e^6)*x^4 + (12*a*c^4*d^5*e^3 - 5*a^2*c^3*d^3*e^5 - 3*a^3*
c^2*d*e^7)*x^3 + (4*a*c^4*d^6*e^2 + 9*a^2*c^3*d^4*e^4 - 9*a^3*c^2*d^2*e^6)*x^2 +
3*(4*a^2*c^3*d^5*e^3 - 3*a^3*c^2*d^3*e^5)*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)))/((a^2*c^
4*d^11 + 4*a^3*c^3*d^9*e^2 + 6*a^4*c^2*d^7*e^4 + 4*a^5*c*d^5*e^6 + a^6*d^3*e^8 +
(a*c^5*d^8*e^3 + 4*a^2*c^4*d^6*e^5 + 6*a^3*c^3*d^4*e^7 + 4*a^4*c^2*d^2*e^9 + a^
5*c*e^11)*x^5 + 3*(a*c^5*d^9*e^2 + 4*a^2*c^4*d^7*e^4 + 6*a^3*c^3*d^5*e^6 + 4*a^4
*c^2*d^3*e^8 + a^5*c*d*e^10)*x^4 + (3*a*c^5*d^10*e + 13*a^2*c^4*d^8*e^3 + 22*a^3
*c^3*d^6*e^5 + 18*a^4*c^2*d^4*e^7 + 7*a^5*c*d^2*e^9 + a^6*e^11)*x^3 + (a*c^5*d^1
1 + 7*a^2*c^4*d^9*e^2 + 18*a^3*c^3*d^7*e^4 + 22*a^4*c^2*d^5*e^6 + 13*a^5*c*d^3*e
^8 + 3*a^6*d*e^10)*x^2 + 3*(a^2*c^4*d^10*e + 4*a^3*c^3*d^8*e^3 + 6*a^4*c^2*d^6*e
^5 + 4*a^5*c*d^4*e^7 + a^6*d^2*e^9)*x)*sqrt(c*d^2 + a*e^2)), 1/6*((24*a*c^3*d^6*
e - 84*a^2*c^2*d^4*e^3 - 5*a^3*c*d^2*e^5 - 2*a^4*e^7 + (6*c^4*d^4*e^3 - 83*a*c^3
*d^2*e^5 + 16*a^2*c^2*e^7)*x^4 + 3*(6*c^4*d^5*e^2 - 63*a*c^3*d^3*e^4 + a^2*c^2*d
*e^6)*x^3 + 2*(9*c^4*d^6*e - 48*a*c^3*d^4*e^3 - 53*a^2*c^2*d^2*e^5 + 4*a^3*c*e^7
)*x^2 + 3*(2*c^4*d^7 + 12*a*c^3*d^5*e^2 - 57*a^2*c^2*d^3*e^4 + 3*a^3*c*d*e^6)*x)
*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) + 15*(4*a^2*c^3*d^6*e^2 - 3*a^3*c^2*d^4*e^
4 + (4*a*c^4*d^3*e^5 - 3*a^2*c^3*d*e^7)*x^5 + 3*(4*a*c^4*d^4*e^4 - 3*a^2*c^3*d^2
*e^6)*x^4 + (12*a*c^4*d^5*e^3 - 5*a^2*c^3*d^3*e^5 - 3*a^3*c^2*d*e^7)*x^3 + (4*a*
c^4*d^6*e^2 + 9*a^2*c^3*d^4*e^4 - 9*a^3*c^2*d^2*e^6)*x^2 + 3*(4*a^2*c^3*d^5*e^3
- 3*a^3*c^2*d^3*e^5)*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))))/((a^2*c^4*d^11 + 4*a^3*c^3*d^9*e^2 + 6*a^4*c^2*d^7*e^4 + 4
*a^5*c*d^5*e^6 + a^6*d^3*e^8 + (a*c^5*d^8*e^3 + 4*a^2*c^4*d^6*e^5 + 6*a^3*c^3*d^
4*e^7 + 4*a^4*c^2*d^2*e^9 + a^5*c*e^11)*x^5 + 3*(a*c^5*d^9*e^2 + 4*a^2*c^4*d^7*e
^4 + 6*a^3*c^3*d^5*e^6 + 4*a^4*c^2*d^3*e^8 + a^5*c*d*e^10)*x^4 + (3*a*c^5*d^10*e
+ 13*a^2*c^4*d^8*e^3 + 22*a^3*c^3*d^6*e^5 + 18*a^4*c^2*d^4*e^7 + 7*a^5*c*d^2*e^
9 + a^6*e^11)*x^3 + (a*c^5*d^11 + 7*a^2*c^4*d^9*e^2 + 18*a^3*c^3*d^7*e^4 + 22*a^
4*c^2*d^5*e^6 + 13*a^5*c*d^3*e^8 + 3*a^6*d*e^10)*x^2 + 3*(a^2*c^4*d^10*e + 4*a^3
*c^3*d^8*e^3 + 6*a^4*c^2*d^6*e^5 + 4*a^5*c*d^4*e^7 + a^6*d^2*e^9)*x)*sqrt(-c*d^2
- a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**4/(c*x**2+a)**(3/2),x)

[Out]

Integral(1/((a + c*x**2)**(3/2)*(d + e*x)**4), x)

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GIAC/XCAS [A]  time = 0.717006, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(3/2)*(e*x + d)^4),x, algorithm="giac")

[Out]

sage0*x