3.572 \(\int \frac{1}{(d+e x)^2 \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=244 \[ \frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 -
4*a*e^2) - c*d*(2*c*d^2 + 7*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a
+ c*x^2]) + (e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)*Sqrt[a + c*x^2])/(3*a^2*(
c*d^2 + a*e^2)^3*(d + e*x)) - (5*c*d*e^4*ArcTanh[(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)

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Rubi [A]  time = 0.641721, antiderivative size = 244, 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 \[ \frac{e \sqrt{a+c x^2} \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )}{3 a^2 (d+e x) \left (a e^2+c d^2\right )^3}-\frac{a e \left (c d^2-4 a e^2\right )-c d x \left (7 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x) \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x) \left (a e^2+c d^2\right )}-\frac{5 c d e^4 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)*(a + c*x^2)^(3/2)) - (a*e*(c*d^2 -
4*a*e^2) - c*d*(2*c*d^2 + 7*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)*Sqrt[a
+ c*x^2]) + (e*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)*Sqrt[a + c*x^2])/(3*a^2*(
c*d^2 + a*e^2)^3*(d + e*x)) - (5*c*d*e^4*ArcTanh[(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)

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Rubi in Sympy [A]  time = 83.4435, size = 221, normalized size = 0.91 \[ - \frac{5 c d e^{4} \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{\left (a e^{2} + c d^{2}\right )^{\frac{7}{2}}} + \frac{a e + c d x}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )} - \frac{e \sqrt{a + c x^{2}} \left (8 a^{2} e^{4} - 9 a c d^{2} e^{2} - 2 c^{2} d^{4}\right )}{3 a^{2} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{3}} + \frac{a e \left (4 a e^{2} - c d^{2}\right ) + c d x \left (7 a e^{2} + 2 c d^{2}\right )}{3 a^{2} \sqrt{a + c x^{2}} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**2/(c*x**2+a)**(5/2),x)

[Out]

-5*c*d*e**4*atanh((a*e - c*d*x)/(sqrt(a + c*x**2)*sqrt(a*e**2 + c*d**2)))/(a*e**
2 + c*d**2)**(7/2) + (a*e + c*d*x)/(3*a*(a + c*x**2)**(3/2)*(d + e*x)*(a*e**2 +
c*d**2)) - e*sqrt(a + c*x**2)*(8*a**2*e**4 - 9*a*c*d**2*e**2 - 2*c**2*d**4)/(3*a
**2*(d + e*x)*(a*e**2 + c*d**2)**3) + (a*e*(4*a*e**2 - c*d**2) + c*d*x*(7*a*e**2
+ 2*c*d**2))/(3*a**2*sqrt(a + c*x**2)*(d + e*x)*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 1.133, size = 222, normalized size = 0.91 \[ \frac{\sqrt{a+c x^2} \left (\frac{c \left (a^2 e^3 (12 d-5 e x)+9 a c d^2 e^2 x+2 c^2 d^4 x\right )}{a^2 \left (a+c x^2\right )}+\frac{c \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a \left (a+c x^2\right )^2}-\frac{3 e^5}{d+e x}\right )}{3 \left (a e^2+c d^2\right )^3}-\frac{5 c d e^4 \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{7/2}}+\frac{5 c d e^4 \log (d+e x)}{\left (a e^2+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^2*(a + c*x^2)^(5/2)),x]

[Out]

(Sqrt[a + c*x^2]*((-3*e^5)/(d + e*x) + (c*(2*c^2*d^4*x + 9*a*c*d^2*e^2*x + a^2*e
^3*(12*d - 5*e*x)))/(a^2*(a + c*x^2)) + (c*(c*d^2 + a*e^2)*(c*d^2*x + a*e*(2*d -
e*x)))/(a*(a + c*x^2)^2)))/(3*(c*d^2 + a*e^2)^3) + (5*c*d*e^4*Log[d + e*x])/(c*
d^2 + a*e^2)^(7/2) - (5*c*d*e^4*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)

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Maple [B]  time = 0.018, size = 667, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^2/(c*x^2+a)^(5/2),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)^(3/2)+5
/3*e*c*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)+5
/3*c^2*d^2/(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)^(3/
2)*x+10/3*c^2*d^2/(a*e^2+c*d^2)^2/a^2/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)
/e^2)^(1/2)*x+5*e^3*c*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)+5*e^2*c^2*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-5*e^3*c*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)*c/a/(c*(d/e+x)^2
-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^2)^(3/2)*x-8/3/(a*e^2+c*d^2)*c/a^2/(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)^(5/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.543307, 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)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

[1/6*(2*(2*a^2*c^2*d^4*e + 14*a^3*c*d^2*e^3 - 3*a^4*e^5 + (2*c^4*d^4*e + 9*a*c^3
*d^2*e^3 - 8*a^2*c^2*e^5)*x^4 + (2*c^4*d^5 + 9*a*c^3*d^3*e^2 + 7*a^2*c^2*d*e^4)*
x^3 + 3*(a*c^3*d^4*e + 7*a^2*c^2*d^2*e^3 - 4*a^3*c*e^5)*x^2 + (3*a*c^3*d^5 + 11*
a^2*c^2*d^3*e^2 + 8*a^3*c*d*e^4)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*x^2 + a) + 15*(a^
2*c^3*d*e^5*x^5 + a^2*c^3*d^2*e^4*x^4 + 2*a^3*c^2*d*e^5*x^3 + 2*a^3*c^2*d^2*e^4*
x^2 + a^4*c*d*e^5*x + a^4*c*d^2*e^4)*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^4*c^3*d^7 + 3
*a^5*c^2*d^5*e^2 + 3*a^6*c*d^3*e^4 + a^7*d*e^6 + (a^2*c^5*d^6*e + 3*a^3*c^4*d^4*
e^3 + 3*a^4*c^3*d^2*e^5 + a^5*c^2*e^7)*x^5 + (a^2*c^5*d^7 + 3*a^3*c^4*d^5*e^2 +
3*a^4*c^3*d^3*e^4 + a^5*c^2*d*e^6)*x^4 + 2*(a^3*c^4*d^6*e + 3*a^4*c^3*d^4*e^3 +
3*a^5*c^2*d^2*e^5 + a^6*c*e^7)*x^3 + 2*(a^3*c^4*d^7 + 3*a^4*c^3*d^5*e^2 + 3*a^5*
c^2*d^3*e^4 + a^6*c*d*e^6)*x^2 + (a^4*c^3*d^6*e + 3*a^5*c^2*d^4*e^3 + 3*a^6*c*d^
2*e^5 + a^7*e^7)*x)*sqrt(c*d^2 + a*e^2)), 1/3*((2*a^2*c^2*d^4*e + 14*a^3*c*d^2*e
^3 - 3*a^4*e^5 + (2*c^4*d^4*e + 9*a*c^3*d^2*e^3 - 8*a^2*c^2*e^5)*x^4 + (2*c^4*d^
5 + 9*a*c^3*d^3*e^2 + 7*a^2*c^2*d*e^4)*x^3 + 3*(a*c^3*d^4*e + 7*a^2*c^2*d^2*e^3
- 4*a^3*c*e^5)*x^2 + (3*a*c^3*d^5 + 11*a^2*c^2*d^3*e^2 + 8*a^3*c*d*e^4)*x)*sqrt(
-c*d^2 - a*e^2)*sqrt(c*x^2 + a) + 15*(a^2*c^3*d*e^5*x^5 + a^2*c^3*d^2*e^4*x^4 +
2*a^3*c^2*d*e^5*x^3 + 2*a^3*c^2*d^2*e^4*x^2 + a^4*c*d*e^5*x + a^4*c*d^2*e^4)*arc
tan(sqrt(-c*d^2 - a*e^2)*(c*d*x - a*e)/((c*d^2 + a*e^2)*sqrt(c*x^2 + a))))/((a^4
*c^3*d^7 + 3*a^5*c^2*d^5*e^2 + 3*a^6*c*d^3*e^4 + a^7*d*e^6 + (a^2*c^5*d^6*e + 3*
a^3*c^4*d^4*e^3 + 3*a^4*c^3*d^2*e^5 + a^5*c^2*e^7)*x^5 + (a^2*c^5*d^7 + 3*a^3*c^
4*d^5*e^2 + 3*a^4*c^3*d^3*e^4 + a^5*c^2*d*e^6)*x^4 + 2*(a^3*c^4*d^6*e + 3*a^4*c^
3*d^4*e^3 + 3*a^5*c^2*d^2*e^5 + a^6*c*e^7)*x^3 + 2*(a^3*c^4*d^7 + 3*a^4*c^3*d^5*
e^2 + 3*a^5*c^2*d^3*e^4 + a^6*c*d*e^6)*x^2 + (a^4*c^3*d^6*e + 3*a^5*c^2*d^4*e^3
+ 3*a^6*c*d^2*e^5 + a^7*e^7)*x)*sqrt(-c*d^2 - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**2/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^2),x, algorithm="giac")

[Out]

integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^2), x)