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

Optimal. Leaf size=398 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

[Out]

(2*Sqrt[d + e*x]*(3*c*d^2 - 5*a*e^2 + 24*c*d*e*x)*Sqrt[a + c*x^2])/(105*c*e) + (
2*e*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(7*c) + (4*Sqrt[-a]*d*(3*c*d^2 - 29*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)])/(105*Sqrt[c]*e^2*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(3*c*d^2 -
5*a*e^2)*(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)])/(105*c^(3/2)*e^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.20775, antiderivative size = 398, 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 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (3 c d^2-5 a e^2\right ) \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 c^{3/2} e^2 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} \left (-5 a e^2+3 c d^2+24 c d e x\right )}{105 c e}+\frac{4 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 c d^2-29 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{105 \sqrt{c} e^2 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \left (a+c x^2\right )^{3/2} \sqrt{d+e x}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(3*c*d^2 - 5*a*e^2 + 24*c*d*e*x)*Sqrt[a + c*x^2])/(105*c*e) + (
2*e*Sqrt[d + e*x]*(a + c*x^2)^(3/2))/(7*c) + (4*Sqrt[-a]*d*(3*c*d^2 - 29*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)])/(105*Sqrt[c]*e^2*Sqrt[(Sqrt[c]
*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(3*c*d^2 -
5*a*e^2)*(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)])/(105*c^(3/2)*e^2*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 169.5, size = 382, normalized size = 0.96 \[ \frac{2 e \left (a + c x^{2}\right )^{\frac{3}{2}} \sqrt{d + e x}}{7 c} - \frac{8 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (\frac{5 a e^{2}}{4} - \frac{3 c d^{2}}{4} - 6 c d e x\right )}{105 c e} - \frac{4 d \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (29 a e^{2} - 3 c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{105 \sqrt{c} e^{2} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} + \frac{4 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) \left (5 a e^{2} - 3 c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{105 c^{\frac{3}{2}} e^{2} \sqrt{a + c x^{2}} \sqrt{d + e x}} \]

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*(a + c*x**2)**(3/2)*sqrt(d + e*x)/(7*c) - 8*sqrt(a + c*x**2)*sqrt(d + e*x)*(
5*a*e**2/4 - 3*c*d**2/4 - 6*c*d*e*x)/(105*c*e) - 4*d*sqrt(-a)*sqrt(1 + c*x**2/a)
*sqrt(d + e*x)*(29*a*e**2 - 3*c*d**2)*elliptic_e(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a
)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(105*sqrt(c)*e**2*sqrt(sqrt(c)*sqr
t(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2)) + 4*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)*(5*a*e**2 - 3*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)))/(105*c**(3/2)*e**2*sqrt(a + c*x**2)*
sqrt(d + e*x))

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Mathematica [C]  time = 5.59002, size = 582, normalized size = 1.46 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (10 a e^2+3 c \left (d^2+8 d e x+5 e^2 x^2\right )\right )}{c e}+\frac{4 \left (\sqrt{a} e (d+e x)^{3/2} \left (-5 i a^{3/2} e^3+27 i \sqrt{a} c d^2 e-29 a \sqrt{c} d e^2+3 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3-3 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2+3 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )-d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-29 a^2 e^2+a c \left (3 d^2-29 e^2 x^2\right )+3 c^2 d^2 x^2\right )\right )}{c e^3 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{105 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(10*a*e^2 + 3*c*(d^2 + 8*d*e*x + 5*e^2*x^2)))/(c*
e) + (4*(-(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(-29*a^2*e^2 + 3*c^2*d^2*x^2 +
a*c*(3*d^2 - 29*e^2*x^2))) + Sqrt[c]*d*((3*I)*c^(3/2)*d^3 - 3*Sqrt[a]*c*d^2*e -
(29*I)*a*Sqrt[c]*d*e^2 + 29*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)*Ellipti
cE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqr
t[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + Sqrt[a]*e*(3*c^(3/2)*d^3 + (27*I)*Sqrt[a]*c
*d^2*e - 29*a*Sqrt[c]*d*e^2 - (5*I)*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)*
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*Sqrt[a]*e)]))/(c*e^3*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[
c]]*(d + e*x))))/(105*Sqrt[a + c*x^2])

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Maple [B]  time = 0.134, size = 1386, normalized size = 3.5 \[ \text{result too large to display} \]

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]

2/105*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(15*x^5*c^3*e^5+10*(-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^2*e^5+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)*Ellip
ticF((-(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^3-6*(-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^2*d^4*e+48*a^2*c*(-(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))*d*e^4+48*a*
c^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))*d^3*e^2-58*a^2*c*(-(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))*d*e^4-52*a*c^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))*d^3*e^2+6*(-(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^3*d^5+39*x^4*c^3*d*e
^4+25*x^3*a*c^2*e^5+27*x^3*c^3*d^2*e^3+49*x^2*a*c^2*d*e^4+3*x^2*c^3*d^3*e^2+10*x
*a^2*c*e^5+27*x*a*c^2*d^2*e^3+10*a^2*c*d*e^4+3*a*c^2*d^3*e^2)/e^3/(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. \[ \int \sqrt{c x^{2} + a}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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

[Out]

Exception raised: RuntimeError