3.1286 \(\int \frac{(b d+2 c d x)^{11/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=179 \[ -18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+36 c d^5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}-\frac{d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac{36}{5} c d^3 (b d+2 c d x)^{5/2} \]

[Out]

36*c*(b^2 - 4*a*c)*d^5*Sqrt[b*d + 2*c*d*x] + (36*c*d^3*(b*d + 2*c*d*x)^(5/2))/5
- (d*(b*d + 2*c*d*x)^(9/2))/(a + b*x + c*x^2) - 18*c*(b^2 - 4*a*c)^(5/4)*d^(11/2
)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 18*c*(b^2 - 4*a*c)
^(5/4)*d^(11/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.393756, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-18 c d^{11/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+36 c d^5 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}-\frac{d (b d+2 c d x)^{9/2}}{a+b x+c x^2}+\frac{36}{5} c d^3 (b d+2 c d x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^2,x]

[Out]

36*c*(b^2 - 4*a*c)*d^5*Sqrt[b*d + 2*c*d*x] + (36*c*d^3*(b*d + 2*c*d*x)^(5/2))/5
- (d*(b*d + 2*c*d*x)^(9/2))/(a + b*x + c*x^2) - 18*c*(b^2 - 4*a*c)^(5/4)*d^(11/2
)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 18*c*(b^2 - 4*a*c)
^(5/4)*d^(11/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi in Sympy [A]  time = 97.1076, size = 180, normalized size = 1.01 \[ - 18 c d^{\frac{11}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 18 c d^{\frac{11}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 36 c d^{5} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} + \frac{36 c d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} - \frac{d \left (b d + 2 c d x\right )^{\frac{9}{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**2,x)

[Out]

-18*c*d**(11/2)*(-4*a*c + b**2)**(5/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))) - 18*c*d**(11/2)*(-4*a*c + b**2)**(5/4)*atanh(sqrt(b*d + 2*c*d
*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 36*c*d**5*(-4*a*c + b**2)*sqrt(b*d + 2*c
*d*x) + 36*c*d**3*(b*d + 2*c*d*x)**(5/2)/5 - d*(b*d + 2*c*d*x)**(9/2)/(a + b*x +
c*x**2)

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Mathematica [A]  time = 0.677475, size = 176, normalized size = 0.98 \[ (d (b+2 c x))^{11/2} \left (\frac{-\frac{5 \left (b^2-4 a c\right )^2}{a+x (b+c x)}-16 c \left (40 a c-11 b^2\right )+64 b c^2 x+64 c^3 x^2}{5 (b+2 c x)^5}-\frac{18 c \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}-\frac{18 c \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{11/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(11/2)/(a + b*x + c*x^2)^2,x]

[Out]

(d*(b + 2*c*x))^(11/2)*((-16*c*(-11*b^2 + 40*a*c) + 64*b*c^2*x + 64*c^3*x^2 - (5
*(b^2 - 4*a*c)^2)/(a + x*(b + c*x)))/(5*(b + 2*c*x)^5) - (18*c*(b^2 - 4*a*c)^(5/
4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(11/2) - (18*c*(b^2
- 4*a*c)^(5/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(11/2))

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Maple [B]  time = 0.022, size = 1090, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(11/2)/(c*x^2+b*x+a)^2,x)

[Out]

16/5*c*d^3*(2*c*d*x+b*d)^(5/2)-128*c^2*d^5*a*(2*c*d*x+b*d)^(1/2)+32*c*d^5*b^2*(2
*c*d*x+b*d)^(1/2)-64*c^3*d^7*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*
c*d^2)*a^2+32*c^2*d^7*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*
a*b^2-4*c*d^7*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*b^4-144*
c^3*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1
/4)*(2*c*d*x+b*d)^(1/2)+1)*a^2+72*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*arct
an(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a*b^2-9*c*d^7/(4*a*
c*d^2-b^2*d^2)^(3/4)*2^(1/2)*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+
b*d)^(1/2)+1)*b^4+72*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*ln((2*c*d*x+b*d+(
4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))/
(2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^
2*d^2)^(1/2)))*a^2-36*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*ln((2*c*d*x+b*d+
(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2))
/(2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b
^2*d^2)^(1/2)))*a*b^2+9/2*c*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*ln((2*c*d*x+b*
d+(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2-b^2*d^2)^(1/2
))/(2*c*d*x+b*d-(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)*2^(1/2)+(4*a*c*d^2
-b^2*d^2)^(1/2)))*b^4+144*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*arctan(2^(1/
2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*a^2-72*c^2*d^7/(4*a*c*d^2-b^
2*d^2)^(3/4)*2^(1/2)*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2
)+1)*a*b^2+9*c*d^7/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*arctan(2^(1/2)/(4*a*c*d^2-b
^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)*b^4

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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((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238645, size = 1037, normalized size = 5.79 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

-1/5*(180*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a
^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)*(c*x^2 + b*x + a)*arctan(-((b^10*c^4 - 20
*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9
)*d^22)^(1/4)/((b^2*c - 4*a*c^2)*sqrt(2*c*d*x + b*d)*d^5 - sqrt(2*(b^4*c^3 - 8*a
*b^2*c^4 + 16*a^2*c^5)*d^11*x + (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^11 + sq
rt((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c
^8 - 1024*a^5*c^9)*d^22)))) - 45*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 6
40*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)*(c*x^2 + b*x + a)*
log(-9*(b^2*c - 4*a*c^2)*sqrt(2*c*d*x + b*d)*d^5 + 9*((b^10*c^4 - 20*a*b^8*c^5 +
160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4
)) + 45*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^3*b^4*c^7 + 1280*a^4
*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)*(c*x^2 + b*x + a)*log(-9*(b^2*c - 4*a*c^2)*
sqrt(2*c*d*x + b*d)*d^5 - 9*((b^10*c^4 - 20*a*b^8*c^5 + 160*a^2*b^6*c^6 - 640*a^
3*b^4*c^7 + 1280*a^4*b^2*c^8 - 1024*a^5*c^9)*d^22)^(1/4)) - (64*c^4*d^5*x^4 + 12
8*b*c^3*d^5*x^3 + 48*(5*b^2*c^2 - 12*a*c^3)*d^5*x^2 + 16*(11*b^3*c - 36*a*b*c^2)
*d^5*x - (5*b^4 - 216*a*b^2*c + 720*a^2*c^2)*d^5)*sqrt(2*c*d*x + b*d))/(c*x^2 +
b*x + a)

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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*d*x+b*d)**(11/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252604, size = 767, normalized size = 4.28 \[ 32 \, \sqrt{2 \, c d x + b d} b^{2} c d^{5} - 128 \, \sqrt{2 \, c d x + b d} a c^{2} d^{5} + \frac{16}{5} \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c d^{3} - \frac{9}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{9}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - 9 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - 9 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) + \frac{4 \,{\left (\sqrt{2 \, c d x + b d} b^{4} c d^{7} - 8 \, \sqrt{2 \, c d x + b d} a b^{2} c^{2} d^{7} + 16 \, \sqrt{2 \, c d x + b d} a^{2} c^{3} d^{7}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(11/2)/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

32*sqrt(2*c*d*x + b*d)*b^2*c*d^5 - 128*sqrt(2*c*d*x + b*d)*a*c^2*d^5 + 16/5*(2*c
*d*x + b*d)^(5/2)*c*d^3 - 9/2*sqrt(2)*(b^2*c*d^5 - 4*a*c^2*d^5)*(-b^2*d^2 + 4*a*
c*d^2)^(1/4)*ln(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sqrt(2*c*d*
x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) + 9/2*sqrt(2)*(b^2*c*d^5 - 4*a*c^2*d^5)*(
-b^2*d^2 + 4*a*c*d^2)^(1/4)*ln(2*c*d*x + b*d - sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1
/4)*sqrt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) - 9*(sqrt(2)*b^2*c*d^5 - 4
*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-b
^2*d^2 + 4*a*c*d^2)^(1/4) + 2*sqrt(2*c*d*x + b*d))/(-b^2*d^2 + 4*a*c*d^2)^(1/4))
- 9*(sqrt(2)*b^2*c*d^5 - 4*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4) - 2*sqrt(2*c*d*x + b*d))/(
-b^2*d^2 + 4*a*c*d^2)^(1/4)) + 4*(sqrt(2*c*d*x + b*d)*b^4*c*d^7 - 8*sqrt(2*c*d*x
+ b*d)*a*b^2*c^2*d^7 + 16*sqrt(2*c*d*x + b*d)*a^2*c^3*d^7)/(b^2*d^2 - 4*a*c*d^2
- (2*c*d*x + b*d)^2)