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

Optimal. Leaf size=382 \[ -\frac{8 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 c d e \sqrt{a+c x^2}}{3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e \sqrt{a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )} \]

[Out]

(-2*e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (8*c*d*e*Sqrt[a + c
*x^2])/(3*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) - (8*Sqrt[-a]*c^(3/2)*d*Sqrt[d + e*x]
*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (
-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x
))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c
]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt
[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(c
*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.916761, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{8 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{8 c d e \sqrt{a+c x^2}}{3 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}-\frac{2 e \sqrt{a+c x^2}}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}+\frac{2 \sqrt{-a} \sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*Sqrt[a + c*x^2])/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (8*c*d*e*Sqrt[a + c
*x^2])/(3*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) - (8*Sqrt[-a]*c^(3/2)*d*Sqrt[d + e*x]
*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (
-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(c*d^2 + a*e^2)^2*Sqrt[(Sqrt[c]*(d + e*x
))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*Sqrt[c]*Sqrt[(Sqrt[c
]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt
[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(c
*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 144.699, size = 357, normalized size = 0.93 \[ - \frac{8 c^{\frac{3}{2}} d \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{2 \sqrt{c} \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )} - \frac{8 c d e \sqrt{a + c x^{2}}}{3 \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{2 e \sqrt{a + c x^{2}}}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*c**(3/2)*d*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*elliptic_e(asin(sqrt(-sq
rt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(3*sqrt(sqrt(c)*
sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)*(a*e**2 + c*d**
2)**2) + 2*sqrt(c)*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sq
rt(-a)))*sqrt(1 + c*x**2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2))
, 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(3*sqrt(a + c*x**2)*sqrt(d + e*x)*(a*e**2 +
c*d**2)) - 8*c*d*e*sqrt(a + c*x**2)/(3*sqrt(d + e*x)*(a*e**2 + c*d**2)**2) - 2*e
*sqrt(a + c*x**2)/(3*(d + e*x)**(3/2)*(a*e**2 + c*d**2))

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Mathematica [C]  time = 4.09501, size = 494, normalized size = 1.29 \[ \frac{2 \left (-e^2 \left (a+c x^2\right ) \left (a e^2+c d (5 d+4 e x)\right )+\frac{c (d+e x) \left (i (d+e x)^{3/2} \left (4 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 d e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}+4 \sqrt{c} d (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{3 e \sqrt{a+c x^2} (d+e x)^{3/2} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(e^2*(a + c*x^2)*(a*e^2 + c*d*(5*d + 4*e*x))) + (c*(d + e*x)*(4*d*e^2*Sqrt[
-d - (I*Sqrt[a]*e)/Sqrt[c]]*(a + c*x^2) + 4*Sqrt[c]*d*((-I)*Sqrt[c]*d + Sqrt[a]*
e)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] -
e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sq
rt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + I*
(3*c*d^2 + (4*I)*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))
/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*Ell
ipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I
*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(3*e
*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)*Sqrt[a + c*x^2])

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Maple [B]  time = 0.061, size = 1904, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/
((-a*c)^(1/2)*e+c*d))^(1/2))*x*a*c*d*e^3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)
*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c
)^(1/2)*e-c*d))^(1/2)+EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c
)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x*a*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-
c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2
))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)+3*EllipticF((-(e*x+d)*c/((-a*c)^(1
/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x*c^2*d^3*
e*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+EllipticF((-(e*x+
d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1
/2))*x*c*d^2*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*(-
a*c)^(1/2)-4*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e
-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x*a*c*d*e^3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))
^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e-c*d))^(1/2)-4*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),
(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x*c^2*d^3*e*(-(e*x+d)*c/((-a
*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x
+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))
^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-
((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c*d^2*e^2+(-a*c)^(1/2)*(-(e*
x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^
(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((
-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*
d*e^3+3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(
1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticF((
-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*
d))^(1/2))*c^2*d^4+(-a*c)^(1/2)*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(
-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
-c*d))^(1/2)*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e
-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c*d^3*e-4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(
1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((
-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((
-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c*d^2*e^2-4*(-(e*x+d)*c/((-a*c
)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(
-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*c^2*d^4-4*x^3*c
^2*d*e^3-x^2*a*c*e^4-5*x^2*c^2*d^2*e^2-4*x*a*c*d*e^3-a^2*e^4-5*a*c*d^2*e^2)/(c*x
^2+a)^(1/2)/(a*e^2+c*d^2)^2/(e*x+d)^(3/2)/e

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((e^2*x^2 + 2*d*e*x + d^2)*sqrt(c*x^2 + a)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError