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

Optimal. Leaf size=406 \[ \frac{e \sqrt{a+c x^2} \left (c d^2-3 a e^2\right )}{a \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2]) + (e*(c*d^2 - 3*
a*e^2)*Sqrt[a + c*x^2])/(a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) - (Sqrt[c]*(c*d^2 -
3*a*e^2)*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)])/(Sqrt[-a]*(c*d^2 + a*
e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (Sq
rt[c]*d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*E
llipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt
[c]*d - a*e)])/(Sqrt[-a]*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.04557, antiderivative size = 406, 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 \[ \frac{e \sqrt{a+c x^2} \left (c d^2-3 a e^2\right )}{a \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{a \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{\sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (c d^2-3 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{\sqrt{-a} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^2 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

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

[Out]

(a*e + c*d*x)/(a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2]) + (e*(c*d^2 - 3*
a*e^2)*Sqrt[a + c*x^2])/(a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) - (Sqrt[c]*(c*d^2 -
3*a*e^2)*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)])/(Sqrt[-a]*(c*d^2 + a*
e^2)^2*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (Sq
rt[c]*d*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/a]*E
llipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt
[c]*d - a*e)])/(Sqrt[-a]*(c*d^2 + a*e^2)*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 161.49, size = 374, normalized size = 0.92 \[ \frac{\sqrt{c} d \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{- a} \sqrt{a + c x^{2}} \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )} + \frac{\sqrt{c} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (3 a e^{2} - c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{\sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )^{2}} - \frac{e \sqrt{a + c x^{2}} \left (3 a e^{2} - c d^{2}\right )}{a \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )^{2}} + \frac{a e + c d x}{a \sqrt{a + c x^{2}} \sqrt{d + e x} \left (a e^{2} + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c)*d*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(1 +
c*x**2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqr
t(c)*d*sqrt(-a)))/(sqrt(-a)*sqrt(a + c*x**2)*sqrt(d + e*x)*(a*e**2 + c*d**2)) +
sqrt(c)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(3*a*e**2 - c*d**2)*elliptic_e(asin(sqr
t(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(sqrt(-a)*s
qrt(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) - e*sqrt(a + c*x**2)*(3*a*e**2 - c*d**2)/(a*sqrt(d + e*x)*(a*
e**2 + c*d**2)**2) + (a*e + c*d*x)/(a*sqrt(a + c*x**2)*sqrt(d + e*x)*(a*e**2 + c
*d**2))

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Mathematica [C]  time = 3.91116, size = 583, normalized size = 1.44 \[ \frac{\sqrt{c} (d+e x)^{3/2} \left (3 a^{3/2} e^3-\sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2+i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (-\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}\right ) \left (-3 a^2 e^2+a c \left (d^2-3 e^2 x^2\right )+c^2 d^2 x^2\right )+e \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (c (d+e x) \left (a e (2 d-e x)+c d^2 x\right )-2 a e^3 \left (a+c x^2\right )\right )+\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (4 i \sqrt{a} \sqrt{c} d e-3 a e^2+c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{a e \sqrt{a+c x^2} \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-3*a^2*e^2 + c^2*d^2*x^2 + a*c*(d^2 - 3
*e^2*x^2))) + e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-2*a*e^3*(a + c*x^2) + c*(d +
e*x)*(c*d^2*x + a*e*(2*d - e*x))) + Sqrt[c]*(I*c^(3/2)*d^3 - Sqrt[a]*c*d^2*e - (
3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*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)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]
*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*Sqrt[c]*e*(c*d^2 + (4*I)*Sqrt[a]*Sqrt[c
]*d*e - 3*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)*EllipticF[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*Sq
rt[a]*e)])/(a*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]
*Sqrt[a + c*x^2])

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Maple [B]  time = 0.083, size = 1167, normalized size = 2.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-3*a^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)*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))*e^4-2*(-(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)*El
lipticE((-(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+(-(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+3*a^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))*e^4+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-(-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-3*x^2*a*c*e^4+x^2*c^2*d^2*e^2+x*a*c*d*e
^3+x*c^2*d^3*e-2*a^2*e^4+2*a*c*d^2*e^2)*(c*x^2+a)^(1/2)*(e*x+d)^(1/2)/e/a/(a*e^2
+c*d^2)^2/(c*e*x^3+c*d*x^2+a*e*x+a*d)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((c*e*x^3 + c*d*x^2 + a*e*x + a*d)*sqrt(c*x^2 + a)*sqrt(e*x + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError