3.2449 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=923 \[ -\frac{2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 e^3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{63 e^5 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

[Out]

(-2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3*d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^
2 + 9*a*b*d*e - 24*a^2*e^2) + 3*c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2)
+ e*(160*c^4*d^4 - 2*b^4*e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e) - b^2*c*e^3*(11*b*
d - 27*a*e) + 3*c^2*e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x
+ c*x^2])/(63*e^5*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(16*c^2*d^3 -
b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d - 4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^2 - 2*c
*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3*(c*d^2 - b*d*e + a*e^2)
*(d + e*x)^(7/2)) - (2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (2*Sqrt[
2]*Sqrt[b^2 - 4*a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^
2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*Sqr
t[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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)^2*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]*(2*c*d - b*e)*(128*c^2*d^2 - b^2*e^2 - 4*c*e*(32*b*d -
33*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*
x^2])

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Rubi [A]  time = 3.35113, antiderivative size = 923, 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 \[ -\frac{2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 e^3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{63 e^5 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3*d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^
2 + 9*a*b*d*e - 24*a^2*e^2) + 3*c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2)
+ e*(160*c^4*d^4 - 2*b^4*e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e) - b^2*c*e^3*(11*b*
d - 27*a*e) + 3*c^2*e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x
+ c*x^2])/(63*e^5*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(16*c^2*d^3 -
b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d - 4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^2 - 2*c
*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3*(c*d^2 - b*d*e + a*e^2)
*(d + e*x)^(7/2)) - (2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (2*Sqrt[
2]*Sqrt[b^2 - 4*a*c]*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^
2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 - 76*a*b*d*e + 28*a^2*e^2))*Sqr
t[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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)^2*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]*(2*c*d - b*e)*(128*c^2*d^2 - b^2*e^2 - 4*c*e*(32*b*d -
33*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*
x^2])

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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((c*x**2+b*x+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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Mathematica [C]  time = 15.2357, size = 8108, normalized size = 8.78 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.231, size = 44994, normalized size = 48.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(11/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(11/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a}}{{\left (e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}\right )} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 +
b*x + a)/((e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 10*d^3*e^2*x^2 + 5*d^4*e*x +
d^5)*sqrt(e*x + d)), 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((c*x**2+b*x+a)**(5/2)/(e*x+d)**(11/2),x)

[Out]

Timed out

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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((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(11/2),x, algorithm="giac")

[Out]

Timed out