3.667 \(\int \frac{(d+e x)^{5/2}}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=359 \[ \frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c}+\frac{16 d e \sqrt{a+c x^2} \sqrt{d+e x}}{15 c} \]

[Out]

(16*d*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(15*c) + (2*e*(d + e*x)^(3/2)*Sqrt[a + c*
x^2])/(5*c) - (2*Sqrt[-a]*(23*c*d^2 - 9*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]*Sq
rt[c]*d - a*e)])/(15*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*
Sqrt[a + c*x^2]) + (16*Sqrt[-a]*d*(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)/S
qrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*c^(3/2)*Sqrt[d + e*
x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 0.979393, antiderivative size = 359, 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 \[ \frac{16 \sqrt{-a} d \sqrt{\frac{c x^2}{a}+1} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (23 c d^2-9 a e^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{15 c^{3/2} \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{2 e \sqrt{a+c x^2} (d+e x)^{3/2}}{5 c}+\frac{16 d e \sqrt{a+c x^2} \sqrt{d+e x}}{15 c} \]

Antiderivative was successfully verified.

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

[Out]

(16*d*e*Sqrt[d + e*x]*Sqrt[a + c*x^2])/(15*c) + (2*e*(d + e*x)^(3/2)*Sqrt[a + c*
x^2])/(5*c) - (2*Sqrt[-a]*(23*c*d^2 - 9*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]*Sq
rt[c]*d - a*e)])/(15*c^(3/2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*
Sqrt[a + c*x^2]) + (16*Sqrt[-a]*d*(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)/S
qrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(15*c^(3/2)*Sqrt[d + e*
x]*Sqrt[a + c*x^2])

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Rubi in Sympy [A]  time = 150.669, size = 338, normalized size = 0.94 \[ \frac{16 d e \sqrt{a + c x^{2}} \sqrt{d + e x}}{15 c} + \frac{2 e \sqrt{a + c x^{2}} \left (d + e x\right )^{\frac{3}{2}}}{5 c} + \frac{16 d \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} \left (a e^{2} + c d^{2}\right ) F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 c^{\frac{3}{2}} \sqrt{a + c x^{2}} \sqrt{d + e x}} + \frac{2 \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (9 a e^{2} - 23 c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{15 c^{\frac{3}{2}} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(c*x**2+a)**(1/2),x)

[Out]

16*d*e*sqrt(a + c*x**2)*sqrt(d + e*x)/(15*c) + 2*e*sqrt(a + c*x**2)*(d + e*x)**(
3/2)/(5*c) + 16*d*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqr
t(-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)))/(15*c**(3/2)*sqrt(a + c*x**
2)*sqrt(d + e*x)) + 2*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(9*a*e**2 - 23*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)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sq
rt(-a)))*sqrt(a + c*x**2))

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Mathematica [C]  time = 5.13705, size = 557, normalized size = 1.55 \[ \frac{\sqrt{d+e x} \left (\frac{2 e \left (a+c x^2\right ) (11 d+3 e x)}{c}+\frac{2 \left (\sqrt{c} (d+e x)^{3/2} \left (9 a^{3/2} e^3-23 \sqrt{a} c d^2 e-17 i a \sqrt{c} d e^2+15 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+\sqrt{c} (d+e x)^{3/2} \left (-9 a^{3/2} e^3+23 \sqrt{a} c d^2 e+9 i a \sqrt{c} d e^2-23 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-9 a^2 e^2+a c \left (23 d^2-9 e^2 x^2\right )+23 c^2 d^2 x^2\right )\right )}{c^2 e (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{15 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((2*e*(11*d + 3*e*x)*(a + c*x^2))/c + (2*(e^2*Sqrt[-d - (I*Sqrt[a
]*e)/Sqrt[c]]*(-9*a^2*e^2 + 23*c^2*d^2*x^2 + a*c*(23*d^2 - 9*e^2*x^2)) + Sqrt[c]
*((-23*I)*c^(3/2)*d^3 + 23*Sqrt[a]*c*d^2*e + (9*I)*a*Sqrt[c]*d*e^2 - 9*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)/S
qrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)] + S
qrt[c]*((15*I)*c^(3/2)*d^3 - 23*Sqrt[a]*c*d^2*e - (17*I)*a*Sqrt[c]*d*e^2 + 9*a^(
3/2)*e^3)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sq
rt[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*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(d + e*x))))/(15*Sqrt[a + c*x^2])

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Maple [B]  time = 0.048, size = 1312, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(c*x^2+a)^(1/2),x)

[Out]

-2/15*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(9*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^4-6*(-(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)*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^2-8*(-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^3-15*(-(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)*Ell
ipticF((-(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^4-8*(-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-9*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+14*(-(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+23*(-(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*x^4*c^2*e^4-14*x^3*c^2*d*e^3-3*x^2*a*c*e^4-11*x^2*c^2*d^2*e^
2-14*x*a*c*d*e^3-11*a*c*d^2*e^2)/c^2/e/(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. \[ \int \frac{{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)/(c*x**2+a)**(1/2),x)

[Out]

Integral((d + e*x)**(5/2)/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)^(5/2)/sqrt(c*x^2 + a),x, algorithm="giac")

[Out]

Exception raised: RuntimeError