3.1572 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=473 \[ \frac{5 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^7 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \sqrt{c} (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{12 e^4 (d+e x)^2}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{8 e^6 (d+e x)}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3} \]

[Out]

(5*(64*c^3*d^3 - b^3*e^3 - 16*c^2*d*e*(5*b*d - 2*a*e) + 12*b*c*e^2*(2*b*d - 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]
)/(8*e^6*(d + e*x)) - (5*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - a*e) + 4*c*e*(2*
c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(12*e^4*(d + e*x)^2) + ((4*c*d - b*e + 2*
c*e*x)*(a + b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) - (5*Sqrt[c]*(2*c*d - b*e)*(
8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(2*e^7) + (5*(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*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])])/(16*e^7*Sqrt[c*d^2 - b*d*e + a*e^2])

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Rubi [A]  time = 1.72502, antiderivative size = 473, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^7 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \sqrt{c} (2 c d-b e) \left (-4 c e (2 b d-a e)+b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{2 e^7}-\frac{5 \left (a+b x+c x^2\right )^{3/2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{12 e^4 (d+e x)^2}+\frac{5 \sqrt{a+b x+c x^2} \left (2 c e x \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right )-16 c^2 d e (5 b d-2 a e)+12 b c e^2 (2 b d-a e)-b^3 e^3+64 c^3 d^3\right )}{8 e^6 (d+e x)}+\frac{\left (a+b x+c x^2\right )^{5/2} (-b e+4 c d+2 c e x)}{3 e^2 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(5*(64*c^3*d^3 - b^3*e^3 - 16*c^2*d*e*(5*b*d - 2*a*e) + 12*b*c*e^2*(2*b*d - 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]
)/(8*e^6*(d + e*x)) - (5*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(3*b*d - a*e) + 4*c*e*(2*
c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(12*e^4*(d + e*x)^2) + ((4*c*d - b*e + 2*
c*e*x)*(a + b*x + c*x^2)^(5/2))/(3*e^2*(d + e*x)^3) - (5*Sqrt[c]*(2*c*d - b*e)*(
8*c^2*d^2 + b^2*e^2 - 4*c*e*(2*b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a
+ b*x + c*x^2])])/(2*e^7) + (5*(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*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])])/(16*e^7*Sqrt[c*d^2 - b*d*e + a*e^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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 3.77, size = 562, normalized size = 1.19 \[ \frac{\frac{15 \log (d+e x) \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{15 \left (16 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-8 b^2 c e^3 (4 b d-3 a e)-128 c^3 d^2 e (2 b d-a e)+b^4 e^4+128 c^4 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}-120 \sqrt{c} (2 c d-b e) \left (4 c e (a e-2 b d)+b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+2 e \sqrt{a+x (b+c x)} \left (\frac{(2 c d-b e) \left (4 c e (41 a e-74 b d)+33 b^2 e^2+296 c^2 d^2\right )}{d+e x}-\frac{2 \left (e (a e-b d)+c d^2\right ) \left (4 c e (3 a e-16 b d)+13 b^2 e^2+64 c^2 d^2\right )}{(d+e x)^2}+8 c \left (2 c e (7 a e-33 b d)+15 b^2 e^2+60 c^2 d^2\right )+\frac{8 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-32 c^2 e x (3 c d-2 b e)+16 c^3 e^2 x^2\right )}{48 e^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(d + e*x)^4,x]

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(8*c*(60*c^2*d^2 + 15*b^2*e^2 + 2*c*e*(-33*b*d + 7*a*
e)) - 32*c^2*e*(3*c*d - 2*b*e)*x + 16*c^3*e^2*x^2 + (8*(2*c*d - b*e)*(c*d^2 + e*
(-(b*d) + a*e))^2)/(d + e*x)^3 - (2*(c*d^2 + e*(-(b*d) + a*e))*(64*c^2*d^2 + 13*
b^2*e^2 + 4*c*e*(-16*b*d + 3*a*e)))/(d + e*x)^2 + ((2*c*d - b*e)*(296*c^2*d^2 +
33*b^2*e^2 + 4*c*e*(-74*b*d + 41*a*e)))/(d + e*x)) + (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))*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] - 120*
Sqrt[c]*(2*c*d - b*e)*(8*c^2*d^2 + b^2*e^2 + 4*c*e*(-2*b*d + a*e))*Log[b + 2*c*x
+ 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] - (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))*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 - b*d*e + a*e^
2]*Sqrt[a + x*(b + c*x)]])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(48*e^7)

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Maple [B]  time = 0.049, size = 28593, normalized size = 60.5 \[ \text{output too large to display} \]

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)^4,x)

[Out]

result too large to display

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)^4,x, algorithm="fricas")

[Out]

Timed out

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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((2*c*x+b)*(c*x**2+b*x+a)**(5/2)/(e*x+d)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 17.7955, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^4,x, algorithm="giac")

[Out]

sage0*x