3.1634 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=592 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{35 e^4}+\frac{2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt{d+e x}} \]

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(35*e^4) + (2*(16*c*d - 7*b*e + 2*c*e*x)*(a
+ b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d
- b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*Sqrt[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)])/(35*c*e^5*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e
+ a*e^2)*(128*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*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)])/(35*c*e^5*Sq
rt[d + e*x]*Sqrt[a + b*x + c*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 1.72585, antiderivative size = 592, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{4 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-5 a e)+27 b^2 e^2+128 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (32 b d-29 a e)+3 b^2 e^2+128 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{35 c e^5 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-4 c e (44 b d-5 a e)+51 b^2 e^2-48 c e x (2 c d-b e)+128 c^2 d^2\right )}{35 e^4}+\frac{2 \left (a+b x+c x^2\right )^{3/2} (-7 b e+16 c d+2 c e x)}{7 e^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(128*c^2*d^2 + 51*b^2*e^2 - 4*c*e*(44*b*d - 5*a*e) - 48*c*e*(2*
c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(35*e^4) + (2*(16*c*d - 7*b*e + 2*c*e*x)*(a
+ b*x + c*x^2)^(3/2))/(7*e^2*Sqrt[d + e*x]) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d
- b*e)*(128*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(32*b*d - 29*a*e))*Sqrt[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)])/(35*c*e^5*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4
*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e
+ a*e^2)*(128*c^2*d^2 + 27*b^2*e^2 - 4*c*e*(32*b*d - 5*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)])/(35*c*e^5*Sq
rt[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((2*c*x+b)*(c*x**2+b*x+a)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [C]  time = 6.67407, size = 5373, normalized size = 9.08 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

_______________________________________________________________________________________

Maple [B]  time = 0.065, size = 6527, normalized size = 11. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x + a)^(3/2)*(2*c*x + b)/(e*x + d)^(3/2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b + 2*c*x)*(a + b*x + c*x**2)**(3/2)/(d + e*x)**(3/2), x)

_______________________________________________________________________________________

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)^(3/2)*(2*c*x + b)/(e*x + d)^(3/2),x, algorithm="giac")

[Out]

Timed out