3.573 \(\int \frac{1}{(d+e x)^3 \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=327 \[ \frac{c d e \sqrt{a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{e \sqrt{a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}} \]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^
2 - 5*a*e^2) - c*d*(2*c*d^2 + 9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*S
qrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[a + c*x^2])/
(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*
a^2*e^4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*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)^(9/2))

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Rubi [A]  time = 1.03474, antiderivative size = 327, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{c d e \sqrt{a+c x^2} \left (-81 a^2 e^4+28 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x) \left (a e^2+c d^2\right )^4}+\frac{e \sqrt{a+c x^2} \left (-15 a^2 e^4+24 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 (d+e x)^2 \left (a e^2+c d^2\right )^3}-\frac{a e \left (2 c d^2-5 a e^2\right )-c d x \left (9 a e^2+2 c d^2\right )}{3 a^2 \sqrt{a+c x^2} (d+e x)^2 \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} (d+e x)^2 \left (a e^2+c d^2\right )}-\frac{5 c e^4 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{2 \left (a e^2+c d^2\right )^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*(d + e*x)^2*(a + c*x^2)^(3/2)) - (a*e*(2*c*d^
2 - 5*a*e^2) - c*d*(2*c*d^2 + 9*a*e^2)*x)/(3*a^2*(c*d^2 + a*e^2)^2*(d + e*x)^2*S
qrt[a + c*x^2]) + (e*(4*c^2*d^4 + 24*a*c*d^2*e^2 - 15*a^2*e^4)*Sqrt[a + c*x^2])/
(6*a^2*(c*d^2 + a*e^2)^3*(d + e*x)^2) + (c*d*e*(4*c^2*d^4 + 28*a*c*d^2*e^2 - 81*
a^2*e^4)*Sqrt[a + c*x^2])/(6*a^2*(c*d^2 + a*e^2)^4*(d + e*x)) - (5*c*e^4*(6*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)^(9/2))

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Rubi in Sympy [A]  time = 136.892, size = 306, normalized size = 0.94 \[ \frac{5 c e^{4} \left (a e^{2} - 6 c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e - c d x}{\sqrt{a + c x^{2}} \sqrt{a e^{2} + c d^{2}}} \right )}}{2 \left (a e^{2} + c d^{2}\right )^{\frac{9}{2}}} + \frac{a e + c d x}{3 a \left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )} - \frac{c d e \sqrt{a + c x^{2}} \left (81 a^{2} e^{4} - 28 a c d^{2} e^{2} - 4 c^{2} d^{4}\right )}{6 a^{2} \left (d + e x\right ) \left (a e^{2} + c d^{2}\right )^{4}} - \frac{e \sqrt{a + c x^{2}} \left (15 a^{2} e^{4} - 24 a c d^{2} e^{2} - 4 c^{2} d^{4}\right )}{6 a^{2} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{3}} + \frac{a e \left (5 a e^{2} - 2 c d^{2}\right ) + c d x \left (9 a e^{2} + 2 c d^{2}\right )}{3 a^{2} \sqrt{a + c x^{2}} \left (d + e x\right )^{2} \left (a e^{2} + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

5*c*e**4*(a*e**2 - 6*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)) + (a*e + c*d*x)/(3*a*(a + c*x**2)**(3/2)
*(d + e*x)**2*(a*e**2 + c*d**2)) - c*d*e*sqrt(a + c*x**2)*(81*a**2*e**4 - 28*a*c
*d**2*e**2 - 4*c**2*d**4)/(6*a**2*(d + e*x)*(a*e**2 + c*d**2)**4) - e*sqrt(a + c
*x**2)*(15*a**2*e**4 - 24*a*c*d**2*e**2 - 4*c**2*d**4)/(6*a**2*(d + e*x)**2*(a*e
**2 + c*d**2)**3) + (a*e*(5*a*e**2 - 2*c*d**2) + c*d*x*(9*a*e**2 + 2*c*d**2))/(3
*a**2*sqrt(a + c*x**2)*(d + e*x)**2*(a*e**2 + c*d**2)**2)

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Mathematica [A]  time = 1.77343, size = 296, normalized size = 0.91 \[ \frac{1}{6} \left (\frac{\sqrt{a+c x^2} \left (\frac{2 c \left (a e^2+c d^2\right ) \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )}{a \left (a+c x^2\right )^2}+\frac{4 c \left (-3 a^3 e^5+3 a^2 c d e^3 (5 d-4 e x)+7 a c^2 d^3 e^2 x+c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}-\frac{3 e^5 \left (a e^2+c d^2\right )}{(d+e x)^2}-\frac{33 c d e^5}{d+e x}\right )}{\left (a e^2+c d^2\right )^4}+\frac{15 c e^4 \left (a e^2-6 c d^2\right ) \log \left (\sqrt{a+c x^2} \sqrt{a e^2+c d^2}+a e-c d x\right )}{\left (a e^2+c d^2\right )^{9/2}}+\frac{15 c e^4 \left (6 c d^2-a e^2\right ) \log (d+e x)}{\left (a e^2+c d^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + c*x^2)^(5/2)),x]

[Out]

((Sqrt[a + c*x^2]*((-3*e^5*(c*d^2 + a*e^2))/(d + e*x)^2 - (33*c*d*e^5)/(d + e*x)
+ (4*c*(-3*a^3*e^5 + c^3*d^5*x + 7*a*c^2*d^3*e^2*x + 3*a^2*c*d*e^3*(5*d - 4*e*x
)))/(a^2*(a + c*x^2)) + (2*c*(c*d^2 + a*e^2)*(-(a^2*e^3) + c^2*d^3*x + 3*a*c*d*e
*(d - e*x)))/(a*(a + c*x^2)^2)))/(c*d^2 + a*e^2)^4 + (15*c*e^4*(6*c*d^2 - a*e^2)
*Log[d + e*x])/(c*d^2 + a*e^2)^(9/2) + (15*c*e^4*(-6*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)^(9/2))/6

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Maple [B]  time = 0.025, size = 1088, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^2+a)^(5/2),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)^(
3/2)-7/2*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)^(3/2)+35/6*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)^(3/2)+35/6*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)^(3/2)*x+35/3*c^3*d^3/(a*e^2+c*d^2)^3/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+35/2*e^3*c^2*d^2/(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*e^2*c^3*d^3/(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^3*c^2*d^2/(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))
-11/2*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)^(3
/2)*x-11*c^2*d/(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-5/6*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)^(3/2)-5/2*e^3/(a*e^2+c*d^2)^3*c/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d^2)/e^
2)^(1/2)-5/2*e^2/(a*e^2+c*d^2)^3*c^2*d/a/(c*(d/e+x)^2-2*c*d/e*(d/e+x)+(a*e^2+c*d
^2)/e^2)^(1/2)*x+5/2*e^3/(a*e^2+c*d^2)^3*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)^(5/2)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.1489, 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)^3),x, algorithm="fricas")

[Out]

[1/12*(2*(6*a^2*c^3*d^6*e + 64*a^3*c^2*d^4*e^3 - 50*a^4*c*d^2*e^5 - 3*a^5*e^7 +
(4*c^5*d^5*e^2 + 28*a*c^4*d^3*e^4 - 81*a^2*c^3*d*e^6)*x^5 + (8*c^5*d^6*e + 56*a*
c^4*d^4*e^3 - 72*a^2*c^3*d^2*e^5 - 15*a^3*c^2*e^7)*x^4 + 2*(2*c^5*d^7 + 17*a*c^4
*d^5*e^2 + 48*a^2*c^3*d^3*e^4 - 72*a^3*c^2*d*e^6)*x^3 + 2*(6*a*c^4*d^6*e + 57*a^
2*c^3*d^4*e^3 - 64*a^3*c^2*d^2*e^5 - 10*a^4*c*e^7)*x^2 + (6*a*c^4*d^7 + 36*a^2*c
^3*d^5*e^2 + 74*a^3*c^2*d^3*e^4 - 61*a^4*c*d*e^6)*x)*sqrt(c*d^2 + a*e^2)*sqrt(c*
x^2 + a) + 15*(6*a^4*c^2*d^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*
e^8)*x^6 + 2*(6*a^2*c^4*d^3*e^5 - a^3*c^3*d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a
^3*c^3*d^2*e^6 - 2*a^4*c^2*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4*c^2*d*e^7)*x^3
+ (12*a^3*c^3*d^4*e^4 + 4*a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^
5 - a^5*c*d*e^7)*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^4*c^4*d^10 + 4*a^5*c^3*d^8*e^2
+ 6*a^6*c^2*d^6*e^4 + 4*a^7*c*d^4*e^6 + a^8*d^2*e^8 + (a^2*c^6*d^8*e^2 + 4*a^3*
c^5*d^6*e^4 + 6*a^4*c^4*d^4*e^6 + 4*a^5*c^3*d^2*e^8 + a^6*c^2*e^10)*x^6 + 2*(a^2
*c^6*d^9*e + 4*a^3*c^5*d^7*e^3 + 6*a^4*c^4*d^5*e^5 + 4*a^5*c^3*d^3*e^7 + a^6*c^2
*d*e^9)*x^5 + (a^2*c^6*d^10 + 6*a^3*c^5*d^8*e^2 + 14*a^4*c^4*d^6*e^4 + 16*a^5*c^
3*d^4*e^6 + 9*a^6*c^2*d^2*e^8 + 2*a^7*c*e^10)*x^4 + 4*(a^3*c^5*d^9*e + 4*a^4*c^4
*d^7*e^3 + 6*a^5*c^3*d^5*e^5 + 4*a^6*c^2*d^3*e^7 + a^7*c*d*e^9)*x^3 + (2*a^3*c^5
*d^10 + 9*a^4*c^4*d^8*e^2 + 16*a^5*c^3*d^6*e^4 + 14*a^6*c^2*d^4*e^6 + 6*a^7*c*d^
2*e^8 + a^8*e^10)*x^2 + 2*(a^4*c^4*d^9*e + 4*a^5*c^3*d^7*e^3 + 6*a^6*c^2*d^5*e^5
+ 4*a^7*c*d^3*e^7 + a^8*d*e^9)*x)*sqrt(c*d^2 + a*e^2)), 1/6*((6*a^2*c^3*d^6*e +
64*a^3*c^2*d^4*e^3 - 50*a^4*c*d^2*e^5 - 3*a^5*e^7 + (4*c^5*d^5*e^2 + 28*a*c^4*d
^3*e^4 - 81*a^2*c^3*d*e^6)*x^5 + (8*c^5*d^6*e + 56*a*c^4*d^4*e^3 - 72*a^2*c^3*d^
2*e^5 - 15*a^3*c^2*e^7)*x^4 + 2*(2*c^5*d^7 + 17*a*c^4*d^5*e^2 + 48*a^2*c^3*d^3*e
^4 - 72*a^3*c^2*d*e^6)*x^3 + 2*(6*a*c^4*d^6*e + 57*a^2*c^3*d^4*e^3 - 64*a^3*c^2*
d^2*e^5 - 10*a^4*c*e^7)*x^2 + (6*a*c^4*d^7 + 36*a^2*c^3*d^5*e^2 + 74*a^3*c^2*d^3
*e^4 - 61*a^4*c*d*e^6)*x)*sqrt(-c*d^2 - a*e^2)*sqrt(c*x^2 + a) + 15*(6*a^4*c^2*d
^4*e^4 - a^5*c*d^2*e^6 + (6*a^2*c^4*d^2*e^6 - a^3*c^3*e^8)*x^6 + 2*(6*a^2*c^4*d^
3*e^5 - a^3*c^3*d*e^7)*x^5 + (6*a^2*c^4*d^4*e^4 + 11*a^3*c^3*d^2*e^6 - 2*a^4*c^2
*e^8)*x^4 + 4*(6*a^3*c^3*d^3*e^5 - a^4*c^2*d*e^7)*x^3 + (12*a^3*c^3*d^4*e^4 + 4*
a^4*c^2*d^2*e^6 - a^5*c*e^8)*x^2 + 2*(6*a^4*c^2*d^3*e^5 - a^5*c*d*e^7)*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^4*c^
4*d^10 + 4*a^5*c^3*d^8*e^2 + 6*a^6*c^2*d^6*e^4 + 4*a^7*c*d^4*e^6 + a^8*d^2*e^8 +
(a^2*c^6*d^8*e^2 + 4*a^3*c^5*d^6*e^4 + 6*a^4*c^4*d^4*e^6 + 4*a^5*c^3*d^2*e^8 +
a^6*c^2*e^10)*x^6 + 2*(a^2*c^6*d^9*e + 4*a^3*c^5*d^7*e^3 + 6*a^4*c^4*d^5*e^5 + 4
*a^5*c^3*d^3*e^7 + a^6*c^2*d*e^9)*x^5 + (a^2*c^6*d^10 + 6*a^3*c^5*d^8*e^2 + 14*a
^4*c^4*d^6*e^4 + 16*a^5*c^3*d^4*e^6 + 9*a^6*c^2*d^2*e^8 + 2*a^7*c*e^10)*x^4 + 4*
(a^3*c^5*d^9*e + 4*a^4*c^4*d^7*e^3 + 6*a^5*c^3*d^5*e^5 + 4*a^6*c^2*d^3*e^7 + a^7
*c*d*e^9)*x^3 + (2*a^3*c^5*d^10 + 9*a^4*c^4*d^8*e^2 + 16*a^5*c^3*d^6*e^4 + 14*a^
6*c^2*d^4*e^6 + 6*a^7*c*d^2*e^8 + a^8*e^10)*x^2 + 2*(a^4*c^4*d^9*e + 4*a^5*c^3*d
^7*e^3 + 6*a^6*c^2*d^5*e^5 + 4*a^7*c*d^3*e^7 + a^8*d*e^9)*x)*sqrt(-c*d^2 - a*e^2
))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**3/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(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)^(5/2)*(e*x + d)^3),x, algorithm="giac")

[Out]

Exception raised: TypeError