3.668 \(\int \frac{(d+e x)^{3/2}}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=317 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x}}{3 c}-\frac{8 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

[Out]

(2*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(3*c) - (8*Sqrt[-a]*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*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqr
t[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*(c*d^2 + a*e^2)*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^(3/2)*Sqrt[
d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.707336, antiderivative size = 317, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 e \sqrt{a+c x^2} \sqrt{d+e x}}{3 c}-\frac{8 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 \sqrt{c} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(3*c) - (8*Sqrt[-a]*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*Sqrt[c]*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqr
t[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*(c*d^2 + a*e^2)*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^(3/2)*Sqrt[
d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 120.524, size = 298, normalized size = 0.94 \[ \frac{2 e \sqrt{a + c x^{2}} \sqrt{d + e x}}{3 c} - \frac{8 d \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 \sqrt{c} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{2 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 c^{\frac{3}{2}} \sqrt{a + c x^{2}} \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 4.25946, size = 445, normalized size = 1.4 \[ \frac{2 \sqrt{d+e x} \left (\frac{i \sqrt{d+e x} \left (4 i \sqrt{a} \sqrt{c} d e-a e^2+3 c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}+\frac{4 d e^2 \left (a+c x^2\right )}{d+e x}+4 i c d \sqrt{d+e x} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \left (a+c x^2\right )\right )}{3 c e \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(e^2*(a + c*x^2) + (4*d*e^2*(a + c*x^2))/(d + e*x) + (4*I)*c*d*
Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*S
qrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*Sqrt[d + e*x]*EllipticE[I*ArcSin
h[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sq
rt[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))]*Sqrt[d + e*x]*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqr
t[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*c*e*Sqrt[a + c*x^2])

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Maple [B]  time = 0.044, size = 978, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError