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3.1009 \(\int \frac{x^{5/2} (A+B x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=347 \[ -\frac{\sqrt{2} \left (-\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \sqrt{x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

[Out]

(2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c^3 - (2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B
*x^(5/2))/(5*c) - (Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b
^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[
2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b - Sqrt[b^2 - 4
*a*c]]) - (Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 + (b^4*B - A*b^3*c - 4
*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 9.09018, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\sqrt{2} \left (-\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (\frac{2 a^2 B c^2+3 a A b c^2-4 a b^2 B c-A b^3 c+b^4 B}{\sqrt{b^2-4 a c}}+a A c^2-2 a b B c-A b^2 c+b^3 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 \sqrt{x} \left (-a B c-A b c+b^2 B\right )}{c^3}-\frac{2 x^{3/2} (b B-A c)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x]

[Out]

(2*(b^2*B - A*b*c - a*B*c)*Sqrt[x])/c^3 - (2*(b*B - A*c)*x^(3/2))/(3*c^2) + (2*B
*x^(5/2))/(5*c) - (Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 - (b^4*B - A*b
^3*c - 4*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[
2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b - Sqrt[b^2 - 4
*a*c]]) - (Sqrt[2]*(b^3*B - A*b^2*c - 2*a*b*B*c + a*A*c^2 + (b^4*B - A*b^3*c - 4
*a*b^2*B*c + 3*a*A*b*c^2 + 2*a^2*B*c^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[
c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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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(x**(5/2)*(B*x+A)/(c*x**2+b*x+a),x)

[Out]

Timed out

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Mathematica [A]  time = 0.82687, size = 422, normalized size = 1.22 \[ -\frac{2 \sqrt{x} \left (a B c+A b c+b^2 (-B)\right )}{c^3}-\frac{\sqrt{2} \left (a c^2 \left (A \sqrt{b^2-4 a c}-2 a B\right )+b^2 c \left (4 a B-A \sqrt{b^2-4 a c}\right )-a b c \left (2 B \sqrt{b^2-4 a c}+3 A c\right )+b^3 \left (B \sqrt{b^2-4 a c}+A c\right )+b^4 (-B)\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{2} \left (a c^2 \left (A \sqrt{b^2-4 a c}+2 a B\right )-b^2 c \left (A \sqrt{b^2-4 a c}+4 a B\right )+a b c \left (3 A c-2 B \sqrt{b^2-4 a c}\right )+b^3 \left (B \sqrt{b^2-4 a c}-A c\right )+b^4 B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{c^{7/2} \sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}}+\frac{2 x^{3/2} (A c-b B)}{3 c^2}+\frac{2 B x^{5/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^(5/2)*(A + B*x))/(a + b*x + c*x^2),x]

[Out]

(-2*(-(b^2*B) + A*b*c + a*B*c)*Sqrt[x])/c^3 + (2*(-(b*B) + A*c)*x^(3/2))/(3*c^2)
 + (2*B*x^(5/2))/(5*c) - (Sqrt[2]*(-(b^4*B) + b^2*c*(4*a*B - A*Sqrt[b^2 - 4*a*c]
) + a*c^2*(-2*a*B + A*Sqrt[b^2 - 4*a*c]) + b^3*(A*c + B*Sqrt[b^2 - 4*a*c]) - a*b
*c*(3*A*c + 2*B*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b - Sq
rt[b^2 - 4*a*c]]])/(c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sq
rt[2]*(b^4*B + a*c^2*(2*a*B + A*Sqrt[b^2 - 4*a*c]) - b^2*c*(4*a*B + A*Sqrt[b^2 -
 4*a*c]) + a*b*c*(3*A*c - 2*B*Sqrt[b^2 - 4*a*c]) + b^3*(-(A*c) + B*Sqrt[b^2 - 4*
a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(c^(7/2)*S
qrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Maple [B]  time = 0.074, size = 1141, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(B*x+A)/(c*x^2+b*x+a),x)

[Out]

2/5*B*x^(5/2)/c+2/3*A*x^(3/2)/c-2/3/c^2*B*x^(3/2)*b-2/c^2*A*x^(1/2)*b-2*a*B*x^(1
/2)/c^2+2/c^3*x^(1/2)*B*b^2+1/c*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan
h(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A-1/c^2*2^(1/2)/((-b+(-
4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)
^(1/2))*A*b^2-3/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A*b+1/c^2/(-4*a*c+
b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/(
(-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A*b^3-2/c^2*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c
)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*B+1/c^3
*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*
c+b^2)^(1/2))*c)^(1/2))*B*b^3-2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(
1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*
B+4/c^2/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x
^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*B-1/c^3/(-4*a*c+b^2)^(1/
2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*
a*c+b^2)^(1/2))*c)^(1/2))*b^4*B-1/c*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A+1/c^2*2^(1/2)/((b+(-
4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(
1/2))*A*b^2-3/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arct
an(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*A*b+1/c^2/(-4*a*c+b^2)^
(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*
a*c+b^2)^(1/2))*c)^(1/2))*A*b^3+2/c^2*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*a
rctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b*B-1/c^3*2^(1/2)/((
b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*
c)^(1/2))*B*b^3-2/c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*
arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*a^2*B+4/c^2/(-4*a*c+b
^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+
(-4*a*c+b^2)^(1/2))*c)^(1/2))*a*b^2*B-1/c^3/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a
*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2
))*b^4*B

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 \,{\left (3 \, B c x^{\frac{5}{2}} - 5 \,{\left (B b - A c\right )} x^{\frac{3}{2}}\right )}}{15 \, c^{2}} - \int \frac{{\left (A b c -{\left (b^{2} - a c\right )} B\right )} x^{\frac{3}{2}} -{\left (B a b - A a c\right )} \sqrt{x}}{c^{3} x^{2} + b c^{2} x + a c^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*B*c*x^(5/2) - 5*(B*b - A*c)*x^(3/2))/c^2 - integrate(((A*b*c - (b^2 - a*
c)*B)*x^(3/2) - (B*a*b - A*a*c)*sqrt(x))/(c^3*x^2 + b*c^2*x + a*c^2), x)

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Fricas [A]  time = 8.41174, size = 10404, normalized size = 29.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2*a^3
*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^2*b^
5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 + A^4*
a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*
B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B
^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*
b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*
a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 +
 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(
5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))*log
(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*a^4*b
 + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*b^3 +
 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 + 24*A
^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)*c^2
- (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b + A*
a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*
a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^
2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3
*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^
4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4
- 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (
37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^
3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^
4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*
a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sqrt((B
^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B
^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4
)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*
b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b
^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A
^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b
^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 -
 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b + 3*A
^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a^3*b^
3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^4 - 3
*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)*c)*s
qrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (7*B^2
*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4 + A^
2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^12 +
A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 1
2*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*
(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^
4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^
3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b
^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 -
 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*c^8))
*log(-sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*B^2*
a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*a^3*
b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b^5 +
 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*b^8)
*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c - (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a^2*b
 + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2
*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A
^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*
A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B
^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)
*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^
3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2
*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*
b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12
*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c + (b^2*c^7 - 4*a*c^8)*sq
rt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7
 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^
2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B
*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*
a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 +
 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*
B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*
c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^4*b
+ 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3*B*a
^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^3*b^
4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*b^6)
*c)*sqrt(x)) + 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4 - (
7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a*b^4
 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^4*b^
12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^
6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6
 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 +
 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 +
28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^
2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*
c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 - 4*a*
c^8))*log(sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 + 28*A*
B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A*B^2*
a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*a^2*b
^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A^2*B*
b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c + (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(2*B*a
^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*
(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b +
54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 +
 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (
46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*
b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9
)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10
 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*
a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3
+ 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8
)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)
*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^
4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A
^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*
B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b
^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*
A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(
b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3*B*a^
4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 11*A^3
*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B^2*a^
3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2*a^2*
b^6)*c)*sqrt(x)) - 15*sqrt(2)*c^3*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5*A^2*a^2*b)*c^4
 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2*b^3 + 12*A*B*a
*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*a*c^8)*sqrt((B^
4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^
4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)
*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b
^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^
6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^
2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^
10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))/(b^2*c^7 -
4*a*c^8))*log(-sqrt(2)*(B^3*b^10 + 4*(A^2*B*a^4 + A^3*a^3*b)*c^6 - (4*B^3*a^5 +
28*A*B^2*a^4*b + 41*A^2*B*a^3*b^2 + 13*A^3*a^2*b^3)*c^5 + (29*B^3*a^4*b^2 + 87*A
*B^2*a^3*b^3 + 58*A^2*B*a^2*b^4 + 7*A^3*a*b^5)*c^4 - (51*B^3*a^3*b^4 + 80*A*B^2*
a^2*b^5 + 24*A^2*B*a*b^6 + A^3*b^7)*c^3 + (35*B^3*a^2*b^6 + 27*A*B^2*a*b^7 + 3*A
^2*B*b^8)*c^2 - (10*B^3*a*b^8 + 3*A*B^2*b^9)*c + (B*b^5*c^7 - 8*A*a^2*c^10 + 6*(
2*B*a^2*b + A*a*b^2)*c^9 - (7*B*a*b^3 + A*b^4)*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8
 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5
*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 + 11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*
b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 + 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^
5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 +
 A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*
B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a
*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15)))*sqrt(-(B^2*b^7 + (4*A*B*a^3 + 5
*A^2*a^2*b)*c^4 - (7*B^2*a^3*b + 18*A*B*a^2*b^2 + 5*A^2*a*b^3)*c^3 + (14*B^2*a^2
*b^3 + 12*A*B*a*b^4 + A^2*b^5)*c^2 - (7*B^2*a*b^5 + 2*A*B*b^6)*c - (b^2*c^7 - 4*
a*c^8)*sqrt((B^4*b^12 + A^4*a^4*c^8 - 2*(A^2*B^2*a^5 + 6*A^3*B*a^4*b + 3*A^4*a^3
*b^2)*c^7 + (B^4*a^6 + 12*A*B^3*a^5*b + 54*A^2*B^2*a^4*b^2 + 52*A^3*B*a^3*b^3 +
11*A^4*a^2*b^4)*c^6 - 2*(6*B^4*a^5*b^2 + 44*A*B^3*a^4*b^3 + 72*A^2*B^2*a^3*b^4 +
 32*A^3*B*a^2*b^5 + 3*A^4*a*b^6)*c^5 + (46*B^4*a^4*b^4 + 160*A*B^3*a^3*b^5 + 132
*A^2*B^2*a^2*b^6 + 28*A^3*B*a*b^7 + A^4*b^8)*c^4 - 2*(31*B^4*a^3*b^6 + 58*A*B^3*
a^2*b^7 + 24*A^2*B^2*a*b^8 + 2*A^3*B*b^9)*c^3 + (37*B^4*a^2*b^8 + 36*A*B^3*a*b^9
 + 6*A^2*B^2*b^10)*c^2 - 2*(5*B^4*a*b^10 + 2*A*B^3*b^11)*c)/(b^2*c^14 - 4*a*c^15
)))/(b^2*c^7 - 4*a*c^8)) + 4*(B^4*a^3*b^6 - A*B^3*a^2*b^7 + A^4*a^4*c^5 - (7*A^3
*B*a^4*b + 3*A^4*a^3*b^2)*c^4 - (B^4*a^6 + 5*A*B^3*a^5*b - 9*A^2*B^2*a^4*b^2 - 1
1*A^3*B*a^3*b^3 - A^4*a^2*b^4)*c^3 + (6*B^4*a^5*b^2 + 2*A*B^3*a^4*b^3 - 12*A^2*B
^2*a^3*b^4 - 3*A^3*B*a^2*b^5)*c^2 - (5*B^4*a^4*b^4 - 3*A*B^3*a^3*b^5 - 3*A^2*B^2
*a^2*b^6)*c)*sqrt(x)) + 4*(3*B*c^2*x^2 + 15*B*b^2 - 15*(B*a + A*b)*c - 5*(B*b*c
- A*c^2)*x)*sqrt(x))/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(B*x+A)/(c*x**2+b*x+a),x)

[Out]

Integral(x**(5/2)*(A + B*x)/(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 121.346, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^(5/2)/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

Done

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