3.2450 \(\int \frac{(d+e x)^{7/2}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}+\frac{12 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c} \]

[Out]

(2*e*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*
x + c*x^2])/(105*c^3) + (12*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2
])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c) + (8*Sqrt[2]*Sqr
t[b^2 - 4*a*c]*(2*c*d - b*e)*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*a*e))*Sq
rt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(
b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c
]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*Sqrt[(c*(d + e*x))/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c
]*(c*d^2 - b*d*e + a*e^2)*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt
[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))
/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)
])/(105*c^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 2.19997, antiderivative size = 600, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 e \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}+\frac{12 e (d+e x)^{3/2} \sqrt{a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac{2 e (d+e x)^{5/2} \sqrt{a+b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*
x + c*x^2])/(105*c^3) + (12*e*(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2
])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c) + (8*Sqrt[2]*Sqr
t[b^2 - 4*a*c]*(2*c*d - b*e)*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*a*e))*Sq
rt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(
b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c
]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*Sqrt[(c*(d + e*x))/(2*c*d -
(b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c
]*(c*d^2 - b*d*e + a*e^2)*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt
[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))
/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)
])/(105*c^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 14.4312, size = 5340, normalized size = 8.9 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

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

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.069, size = 6947, normalized size = 11.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

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+b*x+a)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [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)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

Timed out