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

Optimal. Leaf size=279 \[ -\frac{3 \sqrt{\sqrt{c} d-\sqrt{a} e} \left (2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{\sqrt{a} e+\sqrt{c} d} \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{d+e x} \left (x \left (2 c d^2-a e^2\right )+a d e\right )}{16 a^2 c \left (a-c x^2\right )}+\frac{(d+e x)^{3/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

[Out]

((a*e + c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a - c*x^2)^2) + (3*Sqrt[d + e*x]*(a*d*e
+ (2*c*d^2 - a*e^2)*x))/(16*a^2*c*(a - c*x^2)) - (3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*
(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4)) + (3*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4
*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqr
t[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4))

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Rubi [A]  time = 1.03647, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{3 \sqrt{\sqrt{c} d-\sqrt{a} e} \left (2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{\sqrt{a} e+\sqrt{c} d} \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{7/4}}+\frac{3 \sqrt{d+e x} \left (x \left (2 c d^2-a e^2\right )+a d e\right )}{16 a^2 c \left (a-c x^2\right )}+\frac{(d+e x)^{3/2} (a e+c d x)}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

((a*e + c*d*x)*(d + e*x)^(3/2))/(4*a*c*(a - c*x^2)^2) + (3*Sqrt[d + e*x]*(a*d*e
+ (2*c*d^2 - a*e^2)*x))/(16*a^2*c*(a - c*x^2)) - (3*Sqrt[Sqrt[c]*d - Sqrt[a]*e]*
(4*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[S
qrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4)) + (3*Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(4
*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqr
t[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(7/4))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.717545, size = 299, normalized size = 1.07 \[ \frac{-\frac{3 \left (a^{3/2} e^3-2 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{3 \left (-a^{3/2} e^3+2 \sqrt{a} c d^2 e-3 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{2 \sqrt{a} \sqrt{c} \sqrt{d+e x} \left (a^2 e (7 d+e x)+a c x \left (10 d^2+d e x+3 e^2 x^2\right )-6 c^2 d^2 x^3\right )}{\left (a-c x^2\right )^2}}{32 a^{5/2} c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*Sqrt[c]*Sqrt[d + e*x]*(-6*c^2*d^2*x^3 + a^2*e*(7*d + e*x) + a*c*x*(1
0*d^2 + d*e*x + 3*e^2*x^2)))/(a - c*x^2)^2 - (3*(4*c^(3/2)*d^3 - 2*Sqrt[a]*c*d^2
*e - 3*a*Sqrt[c]*d*e^2 + a^(3/2)*e^3)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d -
Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] + (3*(4*c^(3/2)*d^3 + 2*Sqrt
[a]*c*d^2*e - 3*a*Sqrt[c]*d*e^2 - a^(3/2)*e^3)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/S
qrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^(3/2
))

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Maple [B]  time = 0.106, size = 1004, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)-3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*(e*x+
d)^(7/2)*c*d^2-1/2*e^3/(c*e^2*x^2-a*e^2)^2*d/a*(e*x+d)^(5/2)+9/8*e/(c*e^2*x^2-a*
e^2)^2*d^3/a^2*(e*x+d)^(5/2)*c+1/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)+17/1
6*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*d^2-9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e
*x+d)^(3/2)*d^4+3/8*e^5/(c*e^2*x^2-a*e^2)^2*d/c*(e*x+d)^(1/2)-3/4*e^3/(c*e^2*x^2
-a*e^2)^2*d^3/a*(e*x+d)^(1/2)+3/8*e/(c*e^2*x^2-a*e^2)^2*d^5/a^2*c*(e*x+d)^(1/2)-
9/32*e^9*a^2*c/(a^5*c^3*e^10)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*a
rctanh(e^2*a*c*(e*x+d)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))*d+3/8*e
^7*a*c^2/(a^5*c^3*e^10)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctanh
(e^2*a*c*(e*x+d)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))*d^3-3/32*e^5/
c/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctanh(e^2*a*c*(e*x+d)^(1/2)/(a^2*
d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))+3/16*e^3/a/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^
(1/2))^(1/2)*arctanh(e^2*a*c*(e*x+d)^(1/2)/(a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^
(1/2))*d^2-9/32*e^9*a^2*c/(a^5*c^3*e^10)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1
/2))^(1/2)*arctan(e^2*a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1
/2))*d+3/8*e^7*a*c^2/(a^5*c^3*e^10)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^
(1/2)*arctan(e^2*a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))*
d^3+3/32*e^5/c/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2)*arctan(e^2*a*c*(e*x+d
)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^3*e^10)^(1/2))^(1/2))-3/16*e^3/a/(-a^2*d*e^4*c^2+
(a^5*c^3*e^10)^(1/2))^(1/2)*arctan(e^2*a*c*(e*x+d)^(1/2)/(-a^2*d*e^4*c^2+(a^5*c^
3*e^10)^(1/2))^(1/2))*d^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate((e*x + d)^(5/2)/(c*x^2 - a)^3, x)

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Fricas [A]  time = 0.244056, size = 1389, normalized size = 4.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^(5/2)/(c*x^2 - a)^3,x, algorithm="fricas")

[Out]

1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7))
+ 16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5
- 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2
- a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7)))*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7)) + 16*c^
2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x
^2 + a^4*c)*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7)) + 16*c^2*d^5 - 20*a*c*d^3*e^2 + 5
*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e
*x + d) - 27*(2*a^3*c^2*d*e^6 + (4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7
)))*sqrt((a^5*c^3*sqrt(e^10/(a^5*c^7)) + 16*c^2*d^5 - 20*a*c*d^3*e^2 + 5*a^2*d*e
^4)/(a^5*c^3))) + 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(e^
10/(a^5*c^7)) - 16*c^2*d^5 + 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16
*c^2*d^4*e^5 - 12*a*c*d^2*e^7 + a^2*e^9)*sqrt(e*x + d) + 27*(2*a^3*c^2*d*e^6 - (
4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqrt(e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(e^10/(a^5
*c^7)) - 16*c^2*d^5 + 20*a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) - 3*(a^2*c^3*x^4
- 2*a^3*c^2*x^2 + a^4*c)*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 + 20*
a*c*d^3*e^2 - 5*a^2*d*e^4)/(a^5*c^3))*log(27*(16*c^2*d^4*e^5 - 12*a*c*d^2*e^7 +
a^2*e^9)*sqrt(e*x + d) - 27*(2*a^3*c^2*d*e^6 - (4*a^5*c^6*d^2 - a^6*c^5*e^2)*sqr
t(e^10/(a^5*c^7)))*sqrt(-(a^5*c^3*sqrt(e^10/(a^5*c^7)) - 16*c^2*d^5 + 20*a*c*d^3
*e^2 - 5*a^2*d*e^4)/(a^5*c^3))) + 4*(a*c*d*e*x^2 + 7*a^2*d*e - 3*(2*c^2*d^2 - a*
c*e^2)*x^3 + (10*a*c*d^2 + a^2*e^2)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*c^2*x
^2 + a^4*c)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(e*x + d)^(5/2)/(c*x^2 - a)^3,x, algorithm="giac")

[Out]

Timed out