∖int(A+Bx)(d+ex)3∕2 ____________________________√a+cx2∖,dx

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3.1480 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=388 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) (5 A e+3 B d) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} e \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} (5 A e+3 B d)}{15 c}+\frac{2 B \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c} \]

[Out]

(2*(3*B*d + 5*A*e)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(15*c) + (2*B*(d + e*x)^(3/2)*

Sqrt[a + c*x^2])/(5*c) - (2*Sqrt[-a]*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*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)])/(15*c^(3/2)*e*Sqrt[(Sqrt[c]*(d + e*x

))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*(3*B*d + 5*A*e)*(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)])/(15*c^(3/2)*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A] time = 1.02956, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) (5 A e+3 B d) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} e \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} e \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 \sqrt{a+c x^2} \sqrt{d+e x} (5 A e+3 B d)}{15 c}+\frac{2 B \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(2*(3*B*d + 5*A*e)*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(15*c) + (2*B*(d + e*x)^(3/2)*

Sqrt[a + c*x^2])/(5*c) - (2*Sqrt[-a]*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*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)])/(15*c^(3/2)*e*Sqrt[(Sqrt[c]*(d + e*x

))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) + (2*Sqrt[-a]*(3*B*d + 5*A*e)*(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)])/(15*c^(3/2)*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A] time = 157.642, size = 369, normalized size = 0.95 \[ \frac{2 B \sqrt{a + c x^{2}} \left (d + e x\right )^{\frac{3}{2}}}{5 c} + \frac{2 \sqrt{a + c x^{2}} \sqrt{d + e x} \left (\frac{5 A e}{3} + B d\right )}{5 c} + \frac{2 \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (5 A e + 3 B d\right ) \left (a e^{2} + c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 c^{\frac{3}{2}} e \sqrt{a + c x^{2}} \sqrt{d + e x}} + \frac{2 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (- 20 A c d e + 9 B a e^{2} - 3 B c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 c^{\frac{3}{2}} e \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

2*B*sqrt(a + c*x**2)*(d + e*x)**(3/2)/(5*c) + 2*sqrt(a + c*x**2)*sqrt(d + e*x)*(

5*A*e/3 + B*d)/(5*c) + 2*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c

)*d*sqrt(-a)))*sqrt(1 + c*x**2/a)*(5*A*e + 3*B*d)*(a*e**2 + c*d**2)*elliptic_f(a

sin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(15*

c**(3/2)*e*sqrt(a + c*x**2)*sqrt(d + e*x)) + 2*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(

d + e*x)*(-20*A*c*d*e + 9*B*a*e**2 - 3*B*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)))/(15*c**(3/2)*e*sqrt(sqr

t(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2))

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Mathematica [C] time = 7.33693, size = 550, normalized size = 1.42 \[ \frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) (5 A e+6 B d+3 B e x)}{c}+\frac{2 \left (e^2 \left (a+c x^2\right ) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-9 a B e^2+20 A c d e+3 B c d^2\right )+\sqrt{c} (d+e x)^{3/2} \left (\sqrt{a} e-i \sqrt{c} d\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (-9 a B e^2+20 A c d e+3 B c d^2\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+i \sqrt{c} e (d+e x)^{3/2} \left (\sqrt{c} d+i \sqrt{a} e\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \left (i \sqrt{a} \sqrt{c} (5 A e+3 B d)-9 a B e+15 A c d\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{c^2 e^2 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 \sqrt{a+c x^2}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(Sqrt[d + e*x]*((2*(6*B*d + 5*A*e + 3*B*e*x)*(a + c*x^2))/c + (2*(e^2*Sqrt[-d -

(I*Sqrt[a]*e)/Sqrt[c]]*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*e^2)*(a + c*x^2) + Sqrt[c

]*((-I)*Sqrt[c]*d + Sqrt[a]*e)*(3*B*c*d^2 + 20*A*c*d*e - 9*a*B*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)*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)] + I*Sqrt[c]*e*(Sqrt

[c]*d + I*Sqrt[a]*e)*(15*A*c*d - 9*a*B*e + I*Sqrt[a]*Sqrt[c]*(3*B*d + 5*A*e))*Sq

rt[(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^2*e^

2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])

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Maple [B] time = 0.059, size = 2127, normalized size = 5.5 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)/c^2*(5*A*(-(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^4+5*A*(-(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^2+15*A*(-(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^3+15*A*(-(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^3*e-20*A*(-(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^3-20*A*(-(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*e+3*B*(-(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)*Ellipti

cF((-(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*d*e^3+3*B*(-(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^3*e-9*B*(-(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^4-9

*B*(-(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+9*B*(-(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^2*e^4+6*B*(-(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^2-3*B*(-(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*B*x^4*c^2*e

^4+5*A*x^3*c^2*e^4+9*B*x^3*c^2*d*e^3+5*A*x^2*c^2*d*e^3+3*B*x^2*a*c*e^4+6*B*x^2*c

^2*d^2*e^2+5*A*x*a*c*e^4+9*B*x*a*c*d*e^3+5*A*a*c*d*e^3+6*B*a*c*d^2*e^2)/e^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. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: RuntimeError

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