3.1571 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=464 \[ \frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 \sqrt{c} e^7}-\frac{15 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{4 e^4 (d+e x)}-\frac{15 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{32 e^6}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2} \]

[Out]

(-15*(64*c^3*d^3 - 7*b^3*e^3 + 4*b*c*e^2*(14*b*d - 5*a*e) - 16*c^2*d*e*(7*b*d -
2*a*e) - 2*c*e*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*b*d - a*e))*x)*Sqrt[a + b*x +
c*x^2])/(32*e^6) - (5*(8*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 2*a*e) + c*e*(2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(3/2))/(4*e^4*(d + e*x)) + ((3*c*d - b*e + c*e*x)*(a +
b*x + c*x^2)^(5/2))/(2*e^2*(d + e*x)^2) + (15*(128*c^4*d^4 + b^4*e^4 - 8*b^2*c*
e^3*(4*b*d - 3*a*e) - 128*c^3*d^2*e*(2*b*d - a*e) + 16*c^2*e^2*(10*b^2*d^2 - 8*a
*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(64*S
qrt[c]*e^7) - (15*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*(8*c^2*d^2 + b^2*e^2
- 4*c*e*(2*b*d - a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 -
b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*e^7)

_______________________________________________________________________________________

Rubi [A]  time = 1.78525, antiderivative size = 464, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 \sqrt{c} e^7}-\frac{15 (2 c d-b e) \sqrt{a e^2-b d e+c d^2} \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-c e (7 b d-2 a e)+b^2 e^2+c e x (2 c d-b e)+8 c^2 d^2\right )}{4 e^4 (d+e x)}-\frac{15 \sqrt{a+b x+c x^2} \left (-2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (7 b d-2 a e)+4 b c e^2 (14 b d-5 a e)-7 b^3 e^3+64 c^3 d^3\right )}{32 e^6}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+3 c d+c e x)}{2 e^2 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(-15*(64*c^3*d^3 - 7*b^3*e^3 + 4*b*c*e^2*(14*b*d - 5*a*e) - 16*c^2*d*e*(7*b*d -
2*a*e) - 2*c*e*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*b*d - a*e))*x)*Sqrt[a + b*x +
c*x^2])/(32*e^6) - (5*(8*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 2*a*e) + c*e*(2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(3/2))/(4*e^4*(d + e*x)) + ((3*c*d - b*e + c*e*x)*(a +
b*x + c*x^2)^(5/2))/(2*e^2*(d + e*x)^2) + (15*(128*c^4*d^4 + b^4*e^4 - 8*b^2*c*
e^3*(4*b*d - 3*a*e) - 128*c^3*d^2*e*(2*b*d - a*e) + 16*c^2*e^2*(10*b^2*d^2 - 8*a
*b*d*e + a^2*e^2))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(64*S
qrt[c]*e^7) - (15*(2*c*d - b*e)*Sqrt[c*d^2 - b*d*e + a*e^2]*(8*c^2*d^2 + b^2*e^2
- 4*c*e*(2*b*d - a*e))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 -
b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*e^7)

_______________________________________________________________________________________

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)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 3.55297, size = 590, normalized size = 1.27 \[ \frac{\frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}-\frac{120 (2 c d-b e) \log (d+e x) \left (b^2 e^3 (a e-b d)-4 c^2 d^2 e (4 b d-3 a e)+c e^2 (3 b d-2 a e)^2+8 c^3 d^4\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{120 (2 c d-b e) \left (b^2 e^3 (a e-b d)-4 c^2 d^2 e (4 b d-3 a e)+c e^2 (3 b d-2 a e)^2+8 c^3 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+2 e \sqrt{a+x (b+c x)} \left (2 c e x \left (4 c e (9 a e-32 b d)+37 b^2 e^2+96 c^2 d^2\right )-\frac{8 \left (8 a c e^2+9 b^2 e^2-44 b c d e+44 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )}{d+e x}+\frac{16 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}+260 a b c e^3-448 a c^2 d e^2+49 b^3 e^3-480 b^2 c d e^2+1056 b c^2 d^2 e-8 c^2 e^2 x^2 (8 c d-7 b e)-640 c^3 d^3+16 c^3 e^3 x^3\right )}{64 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x)^3,x]

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(-640*c^3*d^3 + 1056*b*c^2*d^2*e - 480*b^2*c*d*e^2 -
448*a*c^2*d*e^2 + 49*b^3*e^3 + 260*a*b*c*e^3 + 2*c*e*(96*c^2*d^2 + 37*b^2*e^2 +
4*c*e*(-32*b*d + 9*a*e))*x - 8*c^2*e^2*(8*c*d - 7*b*e)*x^2 + 16*c^3*e^3*x^3 + (1
6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (8*(44*c^2*d^2 - 44*
b*c*d*e + 9*b^2*e^2 + 8*a*c*e^2)*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x)) - (120*(
2*c*d - b*e)*(8*c^3*d^4 - 4*c^2*d^2*e*(4*b*d - 3*a*e) + c*e^2*(3*b*d - 2*a*e)^2
+ b^2*e^3*(-(b*d) + a*e))*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] + (15*(12
8*c^4*d^4 + b^4*e^4 - 8*b^2*c*e^3*(4*b*d - 3*a*e) - 128*c^3*d^2*e*(2*b*d - a*e)
+ 16*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[
a + x*(b + c*x)]])/Sqrt[c] + (120*(2*c*d - b*e)*(8*c^3*d^4 - 4*c^2*d^2*e*(4*b*d
- 3*a*e) + c*e^2*(3*b*d - 2*a*e)^2 + b^2*e^3*(-(b*d) + a*e))*Log[-(b*d) + 2*a*e
- 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/Sqr
t[c*d^2 + e*(-(b*d) + a*e)])/(64*e^7)

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 18705, normalized size = 40.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(c*x^2+b*x+a)^(5/2)/(e*x+d)^3,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^(5/2)*(2*c*x + b)/(e*x + d)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError