∖int (d+ex)7∕2 __________________________________∖left(a+cx2 ∖right)5∕2 ∖,dx

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3.682 \(\int \frac{(d+e x)^{7/2}}{\left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=418 \[ -\frac{\sqrt{d+e x} \left (a e \left (5 a e^2+3 c d^2\right )-2 c d x \left (3 a e^2+2 c d^2\right )\right )}{6 a^2 c^2 \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{5/2} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

[Out]

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

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

+ (2*d*(c*d^2 + 2*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqr

t[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(

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

x^2]) - ((c*d^2 + a*e^2)*(4*c*d^2 + 5*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)])/(6*(-a)^(3/2)*c^(5/2)*Sqrt[d

+ e*x]*Sqrt[a + c*x^2])

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Rubi [A] time = 1.12278, antiderivative size = 418, 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 \[ -\frac{\sqrt{d+e x} \left (a e \left (5 a e^2+3 c d^2\right )-2 c d x \left (3 a e^2+2 c d^2\right )\right )}{6 a^2 c^2 \sqrt{a+c x^2}}-\frac{\sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \left (5 a e^2+4 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} c^{5/2} \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{2 d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (2 a e^2+c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{(d+e x)^{5/2} (a e-c d x)}{3 a c \left (a+c x^2\right )^{3/2}} \]

Antiderivative was successfully veriﬁed.

[In] Int[(d + e*x)^(7/2)/(a + c*x^2)^(5/2),x]

[Out]

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

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

+ (2*d*(c*d^2 + 2*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqr

t[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(

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

x^2]) - ((c*d^2 + a*e^2)*(4*c*d^2 + 5*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)])/(6*(-a)^(3/2)*c^(5/2)*Sqrt[d

+ e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A] time = 162.975, size = 393, normalized size = 0.94 \[ \frac{2 d \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (2 a e^{2} + c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{3 c^{\frac{3}{2}} \left (- a\right )^{\frac{3}{2}} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} - \frac{\sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) \left (5 a e^{2} + 4 c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{6 c^{\frac{5}{2}} \left (- a\right )^{\frac{3}{2}} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{\left (d + e x\right )^{\frac{5}{2}} \left (a e - c d x\right )}{3 a c \left (a + c x^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{d + e x} \left (a e \left (5 a e^{2} + 3 c d^{2}\right ) - 2 c d x \left (3 a e^{2} + 2 c d^{2}\right )\right )}{6 a^{2} c^{2} \sqrt{a + c x^{2}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] rubi_integrate((e*x+d)**(7/2)/(c*x**2+a)**(5/2),x)

[Out]

2*d*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(2*a*e**2 + c*d**2)*elliptic_e(asin(sqrt(-s

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

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

*x**2)) - 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 + 4*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)))/(6*c**(5/2)*(-a)**(3/2

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

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

2*c*d**2))/(6*a**2*c**2*sqrt(a + c*x**2))

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Mathematica [C] time = 6.05051, size = 627, normalized size = 1.5 \[ \frac{\sqrt{d+e x} \left (\frac{-5 a^3 e^3+a^2 c e \left (-5 d^2+2 d e x-7 e^2 x^2\right )+a c^2 d x \left (6 d^2+d e x+8 e^2 x^2\right )+4 c^3 d^3 x^3}{a^2 c^2 \left (a+c x^2\right )}-\frac{-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+i \sqrt{a} c d^2 e+8 a \sqrt{c} d e^2+4 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 \sqrt{c} d (d+e x)^{3/2} \left (2 a^{3/2} e^3+\sqrt{a} c d^2 e-2 i a \sqrt{c} d e^2-i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+4 d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (2 a^2 e^2+a c \left (d^2+2 e^2 x^2\right )+c^2 d^2 x^2\right )}{a^2 c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{6 \sqrt{a+c x^2}} \]

Antiderivative was successfully veriﬁed.

[In] Integrate[(d + e*x)^(7/2)/(a + c*x^2)^(5/2),x]

[Out]

(Sqrt[d + e*x]*((-5*a^3*e^3 + 4*c^3*d^3*x^3 + a^2*c*e*(-5*d^2 + 2*d*e*x - 7*e^2*

x^2) + a*c^2*d*x*(6*d^2 + d*e*x + 8*e^2*x^2))/(a^2*c^2*(a + c*x^2)) - (4*d*e^2*S

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

) + 4*Sqrt[c]*d*((-I)*c^(3/2)*d^3 + Sqrt[a]*c*d^2*e - (2*I)*a*Sqrt[c]*d*e^2 + 2*

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*Sqr

t[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a

]*e)] - Sqrt[a]*e*(4*c^(3/2)*d^3 + I*Sqrt[a]*c*d^2*e + 8*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*Sq

rt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[

a]*e)])/(a^2*c^2*e*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(6*Sqrt[a + c*x

^2])

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Maple [B] time = 0.105, size = 2660, normalized size = 6.4 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((e*x+d)^(7/2)/(c*x^2+a)^(5/2),x)

[Out]

-1/6*(4*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^2*c^3*d^4*e*(-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)-8*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^2*a^2*c^2*d*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)-12*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^2*a*c^3*d^3*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)+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))*a^2*c^2*d^3*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)+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))*a^3*c*d*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)-8*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^3*c*d*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)-6*x*a*c^3*d^4*e+9*El

lipticF((-(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^2*a*c^2*d^2*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)-12*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*c^2*d^3*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)+9*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*c*d^2*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)+4*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^2*d^4*e*(-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)+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^2*a^2*c^2*d*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)+5*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^2*a^2*c*e^5*(-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)+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^2*a*c^3*d^3*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)-8*x^4*a*c^3*d*e^4-9*x^3*a*c^3*d^2*e^3+5*x^2*a^

2*c^2*d*e^4-7*x^2*a*c^3*d^3*e^2+3*x*a^2*c^2*d^2*e^3+5*a^3*c*d*e^4+5*a^2*c^2*d^3*

e^2-4*x^4*c^4*d^3*e^2+7*x^3*a^2*c^2*e^5-4*x^3*c^4*d^4*e+5*x*a^3*c*e^5-4*Elliptic

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

a)^(3/2)

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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((e*x + d)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/(c*x^2 + a)^(5/2), x)

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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a}}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((e*x + d)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(e*x + d)/((c^2*x^4 + 2*a

*c*x^2 + a^2)*sqrt(c*x^2 + a)), x)

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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] integrate((e*x+d)**(7/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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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)^(7/2)/(c*x^2 + a)^(5/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError

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