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

Optimal. Leaf size=223 \[ \frac{c d e \sqrt{a+c x^2} \left (2 c d^2-13 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (2 c d^2-3 a e^2\right )}{2 a (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{3 c e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.542868, antiderivative size = 223, 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 \[ \frac{c d e \sqrt{a+c x^2} \left (2 c d^2-13 a e^2\right )}{2 a (d+e x) \left (a e^2+c d^2\right )^3}+\frac{e \sqrt{a+c x^2} \left (2 c d^2-3 a e^2\right )}{2 a (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{3 c e^2 \left (4 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 64.4173, size = 201, normalized size = 0.9 \[ \frac{3 c e^{2} \left (a e^{2} - 4 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} - \frac{c d e \sqrt{a + c x^{2}} \left (13 a e^{2} - 2 c d^{2}\right )}{2 a \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{3}} - \frac{e \sqrt{a + c x^{2}} \left (3 a e^{2} - 2 c d^{2}\right )}{2 a \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{a \sqrt{a + c x^{2}} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c*e**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)))/(2*(a*e**2 + c*d**2)**(7/2)) - c*d*e*sqrt(a + c*x**2)*(13*a*e**2 - 2*
c*d**2)/(2*a*(d + e*x)*(a*e**2 + c*d**2)**3) - e*sqrt(a + c*x**2)*(3*a*e**2 - 2*
c*d**2)/(2*a*(d + e*x)**2*(a*e**2 + c*d**2)**2) + (a*e + c*d*x)/(a*sqrt(a + c*x*
*2)*(d + e*x)**2*(a*e**2 + c*d**2))

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Mathematica [A]  time = 0.681661, size = 240, normalized size = 1.08 \[ \frac{1}{2} \left (\frac{-a^3 e^5-a^2 c e^3 \left (10 d^2+11 d e x+3 e^2 x^2\right )+a c^2 d e \left (6 d^3+6 d^2 e x-14 d e^2 x^2-13 e^3 x^3\right )+2 c^3 d^3 x (d+e x)^2}{a \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^3}+\frac{3 c e^2 \left (a e^2-4 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{3 c e^2 \left (4 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-(a^3*e^5) + 2*c^3*d^3*x*(d + e*x)^2 - a^2*c*e^3*(10*d^2 + 11*d*e*x + 3*e^2*x^
2) + a*c^2*d*e*(6*d^3 + 6*d^2*e*x - 14*d*e^2*x^2 - 13*e^3*x^3))/(a*(c*d^2 + a*e^
2)^3*(d + e*x)^2*Sqrt[a + c*x^2]) + (3*c*e^2*(4*c*d^2 - a*e^2)*Log[d + e*x])/(c*
d^2 + a*e^2)^(7/2) + (3*c*e^2*(-4*c*d^2 + 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)^(7/2))/2

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Maple [B]  time = 0.02, size = 681, 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)^3/(c*x^2+a)^(3/2),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)^(
1/2)-5/2*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)^(1/2)+15/2*e*c^2*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)+15/2*c^3*d^3/(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^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))-13/2*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-3/2*e/(a*e^2+c*d^2)^2*c
/(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)^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))

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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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.618534, 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)^3),x, algorithm="fricas")

[Out]

[1/4*(2*(6*a*c^2*d^4*e - 10*a^2*c*d^2*e^3 - a^3*e^5 + (2*c^3*d^3*e^2 - 13*a*c^2*
d*e^4)*x^3 + (4*c^3*d^4*e - 14*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + (2*c^3*d^5 + 6
*a*c^2*d^3*e^2 - 11*a^2*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 3*(4*a
^2*c^2*d^4*e^2 - a^3*c*d^2*e^4 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 2*(4*a*c^
3*d^3*e^3 - a^2*c^2*d*e^5)*x^3 + (4*a*c^3*d^4*e^2 + 3*a^2*c^2*d^2*e^4 - a^3*c*e^
6)*x^2 + 2*(4*a^2*c^2*d^3*e^3 - a^3*c*d*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^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3
*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 + a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^2*c
^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*d*e^7)*x^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e
^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a
^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*d*e^7)*x)*sqrt(c*d^2 + a*e^2)), 1/2*((6*a
*c^2*d^4*e - 10*a^2*c*d^2*e^3 - a^3*e^5 + (2*c^3*d^3*e^2 - 13*a*c^2*d*e^4)*x^3 +
(4*c^3*d^4*e - 14*a*c^2*d^2*e^3 - 3*a^2*c*e^5)*x^2 + (2*c^3*d^5 + 6*a*c^2*d^3*e
^2 - 11*a^2*c*d*e^4)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) + 3*(4*a^2*c^2*d^4*
e^2 - a^3*c*d^2*e^4 + (4*a*c^3*d^2*e^4 - a^2*c^2*e^6)*x^4 + 2*(4*a*c^3*d^3*e^3 -
a^2*c^2*d*e^5)*x^3 + (4*a*c^3*d^4*e^2 + 3*a^2*c^2*d^2*e^4 - a^3*c*e^6)*x^2 + 2*
(4*a^2*c^2*d^3*e^3 - a^3*c*d*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^3*d^8 + 3*a^3*c^2*d^6*e^2 + 3*a^4*c*d
^4*e^4 + a^5*d^2*e^6 + (a*c^4*d^6*e^2 + 3*a^2*c^3*d^4*e^4 + 3*a^3*c^2*d^2*e^6 +
a^4*c*e^8)*x^4 + 2*(a*c^4*d^7*e + 3*a^2*c^3*d^5*e^3 + 3*a^3*c^2*d^3*e^5 + a^4*c*
d*e^7)*x^3 + (a*c^4*d^8 + 4*a^2*c^3*d^6*e^2 + 6*a^3*c^2*d^4*e^4 + 4*a^4*c*d^2*e^
6 + a^5*e^8)*x^2 + 2*(a^2*c^3*d^7*e + 3*a^3*c^2*d^5*e^3 + 3*a^4*c*d^3*e^5 + a^5*
d*e^7)*x)*sqrt(-c*d^2 - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError