3.2351 \(\int \frac{\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=337 \[ \frac{5 \sqrt{c} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 e^6}-\frac{5 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^6 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \sqrt{a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{8 e^5 (d+e x)}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{12 e^3 (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]

[Out]

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

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Rubi [A]  time = 1.07056, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{5 \sqrt{c} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 e^6}-\frac{5 (2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{16 e^6 \sqrt{a e^2-b d e+c d^2}}-\frac{5 \sqrt{a+b x+c x^2} \left (-4 c e (3 b d-a e)+b^2 e^2+4 c e x (2 c d-b e)+16 c^2 d^2\right )}{8 e^5 (d+e x)}+\frac{5 \left (a+b x+c x^2\right )^{3/2} (-b e+4 c d+2 c e x)}{12 e^3 (d+e x)^2}-\frac{\left (a+b x+c x^2\right )^{5/2}}{3 e (d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.5898, size = 385, normalized size = 1.14 \[ \frac{2 e \sqrt{a+x (b+c x)} \left (\frac{4 c e (47 b d-14 a e)-33 b^2 e^2-188 c^2 d^2}{d+e x}+\frac{26 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{(d+e x)^2}-\frac{8 \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}+6 c (9 b e-16 c d)+12 c^2 e x\right )+\frac{15 (b e-2 c d) \log (d+e x) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right )}{\sqrt{e (a e-b d)+c d^2}}+30 \sqrt{c} \left (4 c e (a e-4 b d)+3 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+\frac{15 (2 c d-b e) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}}{48 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(2*e*Sqrt[a + x*(b + c*x)]*(6*c*(-16*c*d + 9*b*e) + 12*c^2*e*x - (8*(c*d^2 + e*(
-(b*d) + a*e))^2)/(d + e*x)^3 + (26*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(d
+ e*x)^2 + (-188*c^2*d^2 - 33*b^2*e^2 + 4*c*e*(47*b*d - 14*a*e))/(d + e*x)) + (
15*(-2*c*d + b*e)*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*a*e))*Log[d + e*x])/
Sqrt[c*d^2 + e*(-(b*d) + a*e)] + 30*Sqrt[c]*(16*c^2*d^2 + 3*b^2*e^2 + 4*c*e*(-4*
b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]] + (15*(2*c*d - b*e)
*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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)]])/Sqrt[c*d^2 + e*
(-(b*d) + a*e)])/(48*e^6)

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Maple [B]  time = 0.033, size = 18718, normalized size = 55.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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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)/(e*x + d)^4,x, algorithm="giac")

[Out]

Exception raised: TypeError