3.1562 \(\int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^2} \, dx\)
Optimal. Leaf size=303 \[ -\frac{(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 \sqrt{c} e^5}+\frac{\sqrt{a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^4}+\frac{\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)} \]
[Out]
((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a +
b*x + c*x^2])/(2*e^4) + ((8*c*d - 3*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(3*
e^2*(d + e*x)) - ((2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*A
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(4*Sqrt[c]*e^5) + (Sqrt[c
*d^2 - b*d*e + a*e^2]*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*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
])])/(2*e^5)
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Rubi [A] time = 1.05619, antiderivative size = 303, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214
\[ -\frac{(2 c d-b e) \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{4 \sqrt{c} e^5}+\frac{\sqrt{a e^2-b d e+c d^2} \left (-4 c e (4 b d-a e)+3 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 e^5}+\frac{\sqrt{a+b x+c x^2} \left (-4 c e (5 b d-a e)+5 b^2 e^2-4 c e x (2 c d-b e)+16 c^2 d^2\right )}{2 e^4}+\frac{\left (a+b x+c x^2\right )^{3/2} (-3 b e+8 c d+2 c e x)}{3 e^2 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^2,x]
[Out]
((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(5*b*d - a*e) - 4*c*e*(2*c*d - b*e)*x)*Sqrt[a +
b*x + c*x^2])/(2*e^4) + ((8*c*d - 3*b*e + 2*c*e*x)*(a + b*x + c*x^2)^(3/2))/(3*
e^2*(d + e*x)) - ((2*c*d - b*e)*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*A
rcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(4*Sqrt[c]*e^5) + (Sqrt[c
*d^2 - b*d*e + a*e^2]*(16*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(4*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
])])/(2*e^5)
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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)**(3/2)/(e*x+d)**2,x)
[Out]
Timed out
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Mathematica [A] time = 1.11711, size = 442, normalized size = 1.46 \[ \frac{\frac{6 \log (d+e x) \left (c e^2 \left (4 a^2 e^2-20 a b d e+19 b^2 d^2\right )+3 b^2 e^3 (a e-b d)-4 c^2 d^2 e (8 b d-5 a e)+16 c^3 d^4\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{6 \left (c e^2 \left (4 a^2 e^2-20 a b d e+19 b^2 d^2\right )+3 b^2 e^3 (a e-b d)-4 c^2 d^2 e (8 b d-5 a e)+16 c^3 d^4\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}-\frac{3 (2 c d-b e) \left (4 c e (3 a e-4 b d)+b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{\sqrt{c}}+\frac{2 e \sqrt{a+x (b+c x)} \left (2 c e \left (2 a e (7 d+4 e x)+b \left (-30 d^2-16 d e x+5 e^2 x^2\right )\right )+3 b e^2 (-2 a e+5 b d+3 b e x)+4 c^2 \left (12 d^3+6 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{d+e x}}{12 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^(3/2))/(d + e*x)^2,x]
[Out]
((2*e*Sqrt[a + x*(b + c*x)]*(3*b*e^2*(5*b*d - 2*a*e + 3*b*e*x) + 4*c^2*(12*d^3 +
6*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3) + 2*c*e*(2*a*e*(7*d + 4*e*x) + b*(-30*d^2 -
16*d*e*x + 5*e^2*x^2))))/(d + e*x) + (6*(16*c^3*d^4 - 4*c^2*d^2*e*(8*b*d - 5*a*e
) + 3*b^2*e^3*(-(b*d) + a*e) + c*e^2*(19*b^2*d^2 - 20*a*b*d*e + 4*a^2*e^2))*Log[
d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] - (3*(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 + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/Sq
rt[c] - (6*(16*c^3*d^4 - 4*c^2*d^2*e*(8*b*d - 5*a*e) + 3*b^2*e^3*(-(b*d) + a*e)
+ c*e^2*(19*b^2*d^2 - 20*a*b*d*e + 4*a^2*e^2))*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)])/(12*e^5)
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Maple [B] time = 0.02, size = 6898, normalized size = 22.8 \[ \text{output too large to display} \]
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)^2,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)^(3/2)*(2*c*x + b)/(e*x + d)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 140.386, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)^2,x, algorithm="fricas")
[Out]
[1/24*(6*(16*c^2*d^3 - 16*b*c*d^2*e + (3*b^2 + 4*a*c)*d*e^2 + (16*c^2*d^2*e - 16
*b*c*d*e^2 + (3*b^2 + 4*a*c)*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*log((8*
a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)
*e^2)*x^2 - 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (
2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 +
2*d*e*x + d^2)) + 4*(4*c^2*e^4*x^3 + 48*c^2*d^3*e - 60*b*c*d^2*e^2 - 6*a*b*e^4 +
(15*b^2 + 28*a*c)*d*e^3 - 2*(4*c^2*d*e^3 - 5*b*c*e^4)*x^2 + (24*c^2*d^2*e^2 - 3
2*b*c*d*e^3 + (9*b^2 + 16*a*c)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 3*(32*c^3
*d^4 - 48*b*c^2*d^3*e + 6*(3*b^2*c + 4*a*c^2)*d^2*e^2 - (b^3 + 12*a*b*c)*d*e^3 +
(32*c^3*d^3*e - 48*b*c^2*d^2*e^2 + 6*(3*b^2*c + 4*a*c^2)*d*e^3 - (b^3 + 12*a*b*
c)*e^4)*x)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x +
b^2 + 4*a*c)*sqrt(c)))/((e^6*x + d*e^5)*sqrt(c)), 1/12*(3*(16*c^2*d^3 - 16*b*c*
d^2*e + (3*b^2 + 4*a*c)*d*e^2 + (16*c^2*d^2*e - 16*b*c*d*e^2 + (3*b^2 + 4*a*c)*e
^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(-c)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 +
4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*
d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^
2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(4*c^2*e^
4*x^3 + 48*c^2*d^3*e - 60*b*c*d^2*e^2 - 6*a*b*e^4 + (15*b^2 + 28*a*c)*d*e^3 - 2*
(4*c^2*d*e^3 - 5*b*c*e^4)*x^2 + (24*c^2*d^2*e^2 - 32*b*c*d*e^3 + (9*b^2 + 16*a*c
)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(32*c^3*d^4 - 48*b*c^2*d^3*e + 6*(3
*b^2*c + 4*a*c^2)*d^2*e^2 - (b^3 + 12*a*b*c)*d*e^3 + (32*c^3*d^3*e - 48*b*c^2*d^
2*e^2 + 6*(3*b^2*c + 4*a*c^2)*d*e^3 - (b^3 + 12*a*b*c)*e^4)*x)*arctan(1/2*(2*c*x
+ b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((e^6*x + d*e^5)*sqrt(-c)), -1/24*(12
*(16*c^2*d^3 - 16*b*c*d^2*e + (3*b^2 + 4*a*c)*d*e^2 + (16*c^2*d^2*e - 16*b*c*d*e
^2 + (3*b^2 + 4*a*c)*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c)*arctan(-1/2*(b
*d - 2*a*e + (2*c*d - b*e)*x)/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a
))) - 4*(4*c^2*e^4*x^3 + 48*c^2*d^3*e - 60*b*c*d^2*e^2 - 6*a*b*e^4 + (15*b^2 + 2
8*a*c)*d*e^3 - 2*(4*c^2*d*e^3 - 5*b*c*e^4)*x^2 + (24*c^2*d^2*e^2 - 32*b*c*d*e^3
+ (9*b^2 + 16*a*c)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(32*c^3*d^4 - 48*b*
c^2*d^3*e + 6*(3*b^2*c + 4*a*c^2)*d^2*e^2 - (b^3 + 12*a*b*c)*d*e^3 + (32*c^3*d^3
*e - 48*b*c^2*d^2*e^2 + 6*(3*b^2*c + 4*a*c^2)*d*e^3 - (b^3 + 12*a*b*c)*e^4)*x)*l
og(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c
)*sqrt(c)))/((e^6*x + d*e^5)*sqrt(c)), -1/12*(6*(16*c^2*d^3 - 16*b*c*d^2*e + (3*
b^2 + 4*a*c)*d*e^2 + (16*c^2*d^2*e - 16*b*c*d*e^2 + (3*b^2 + 4*a*c)*e^3)*x)*sqrt
(-c*d^2 + b*d*e - a*e^2)*sqrt(-c)*arctan(-1/2*(b*d - 2*a*e + (2*c*d - b*e)*x)/(s
qrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a))) - 2*(4*c^2*e^4*x^3 + 48*c^2*
d^3*e - 60*b*c*d^2*e^2 - 6*a*b*e^4 + (15*b^2 + 28*a*c)*d*e^3 - 2*(4*c^2*d*e^3 -
5*b*c*e^4)*x^2 + (24*c^2*d^2*e^2 - 32*b*c*d*e^3 + (9*b^2 + 16*a*c)*e^4)*x)*sqrt(
c*x^2 + b*x + a)*sqrt(-c) + 3*(32*c^3*d^4 - 48*b*c^2*d^3*e + 6*(3*b^2*c + 4*a*c^
2)*d^2*e^2 - (b^3 + 12*a*b*c)*d*e^3 + (32*c^3*d^3*e - 48*b*c^2*d^2*e^2 + 6*(3*b^
2*c + 4*a*c^2)*d*e^3 - (b^3 + 12*a*b*c)*e^4)*x)*arctan(1/2*(2*c*x + b)*sqrt(-c)/
(sqrt(c*x^2 + b*x + a)*c)))/((e^6*x + d*e^5)*sqrt(-c))]
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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)**(3/2)/(e*x+d)**2,x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]
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)^2,x, algorithm="giac")
[Out]
undef