3.1285 \(\int \frac{(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=181 \[ 22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{44}{3} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2} \]

[Out]

(44*c*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2))/3 + (44*c*d^3*(b*d + 2*c*d*x)^(7/
2))/7 - (d*(b*d + 2*c*d*x)^(11/2))/(a + b*x + c*x^2) + 22*c*(b^2 - 4*a*c)^(7/4)*
d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 22*c*(b^2 -
4*a*c)^(7/4)*d^(13/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])
]

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Rubi [A]  time = 0.412921, antiderivative size = 181, 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 \[ 22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-22 c d^{13/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{44}{3} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}-\frac{d (b d+2 c d x)^{11/2}}{a+b x+c x^2}+\frac{44}{7} c d^3 (b d+2 c d x)^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^2,x]

[Out]

(44*c*(b^2 - 4*a*c)*d^5*(b*d + 2*c*d*x)^(3/2))/3 + (44*c*d^3*(b*d + 2*c*d*x)^(7/
2))/7 - (d*(b*d + 2*c*d*x)^(11/2))/(a + b*x + c*x^2) + 22*c*(b^2 - 4*a*c)^(7/4)*
d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 22*c*(b^2 -
4*a*c)^(7/4)*d^(13/2)*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 = 96.1226, size = 182, normalized size = 1.01 \[ 22 c d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 22 c d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + \frac{44 c d^{5} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} + \frac{44 c d^{3} \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7} - \frac{d \left (b d + 2 c d x\right )^{\frac{11}{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(13/2)/(c*x**2+b*x+a)**2,x)

[Out]

22*c*d**(13/2)*(-4*a*c + b**2)**(7/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))) - 22*c*d**(13/2)*(-4*a*c + b**2)**(7/4)*atanh(sqrt(b*d + 2*c*d*
x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 44*c*d**5*(-4*a*c + b**2)*(b*d + 2*c*d*x)
**(3/2)/3 + 44*c*d**3*(b*d + 2*c*d*x)**(7/2)/7 - d*(b*d + 2*c*d*x)**(11/2)/(a +
b*x + c*x**2)

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Mathematica [A]  time = 0.704128, size = 207, normalized size = 1.14 \[ (d (b+2 c x))^{13/2} \left (\frac{16 c^2 \left (-77 a^2-44 a c x^2+12 c^2 x^4\right )+8 b^2 c \left (55 a+58 c x^2\right )+64 b c^2 x \left (6 c x^2-11 a\right )-21 b^4+272 b^3 c x}{21 (b+2 c x)^5 (a+x (b+c x))}+\frac{22 c \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}-\frac{22 c \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^2,x]

[Out]

(d*(b + 2*c*x))^(13/2)*((-21*b^4 + 272*b^3*c*x + 64*b*c^2*x*(-11*a + 6*c*x^2) +
8*b^2*c*(55*a + 58*c*x^2) + 16*c^2*(-77*a^2 - 44*a*c*x^2 + 12*c^2*x^4))/(21*(b +
2*c*x)^5*(a + x*(b + c*x))) + (22*c*(b^2 - 4*a*c)^(7/4)*ArcTan[Sqrt[b + 2*c*x]/
(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(13/2) - (22*c*(b^2 - 4*a*c)^(7/4)*ArcTanh[Sqr
t[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(13/2))

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Maple [B]  time = 0.021, size = 1090, normalized size = 6. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(13/2)/(c*x^2+b*x+a)^2,x)

[Out]

16/7*c*d^3*(2*c*d*x+b*d)^(7/2)-128/3*c^2*d^5*(2*c*d*x+b*d)^(3/2)*a+32/3*c*d^5*b^
2*(2*c*d*x+b*d)^(3/2)-64*c^3*d^7*(2*c*d*x+b*d)^(3/2)/(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)^(3/2)/(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)^(3/2)/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*b^4+
88*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a^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)))
+176*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a^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)-176*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)
*a^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)-44*c^2*d^7
/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a*b^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)))-88*c^2*
d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a*b^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)+88*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a*b^2*a
rctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)+11/2*c*d^7/(4*a*
c*d^2-b^2*d^2)^(1/4)*2^(1/2)*b^4*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)))+11*c*d^7/(4*a*c
*d^2-b^2*d^2)^(1/4)*2^(1/2)*b^4*arctan(2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*
x+b*d)^(1/2)+1)-11*c*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*b^4*arctan(-2^(1/2)/(
4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)

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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)^(13/2)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250409, size = 1828, normalized size = 10.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(13/2)/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

1/21*(924*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 896
0*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(
1/4)*(c*x^2 + b*x + a)*arctan(-((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 - 2
240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 - 16
384*a^7*c^11)*d^26)^(3/4)/((b^10*c^3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*
b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)*sqrt(2*c*d*x + b*d)*d^19 - sqrt(2*(b^
20*c^7 - 40*a*b^18*c^8 + 720*a^2*b^16*c^9 - 7680*a^3*b^14*c^10 + 53760*a^4*b^12*
c^11 - 258048*a^5*b^10*c^12 + 860160*a^6*b^8*c^13 - 1966080*a^7*b^6*c^14 + 29491
20*a^8*b^4*c^15 - 2621440*a^9*b^2*c^16 + 1048576*a^10*c^17)*d^39*x + (b^21*c^6 -
40*a*b^19*c^7 + 720*a^2*b^17*c^8 - 7680*a^3*b^15*c^9 + 53760*a^4*b^13*c^10 - 25
8048*a^5*b^11*c^11 + 860160*a^6*b^9*c^12 - 1966080*a^7*b^7*c^13 + 2949120*a^8*b^
5*c^14 - 2621440*a^9*b^3*c^15 + 1048576*a^10*b*c^16)*d^39 + sqrt((b^14*c^4 - 28*
a*b^12*c^5 + 336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*
b^4*c^9 + 28672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)*(b^14*c^4 - 28*a*b^12*c^5 +
336*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28
672*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26))) + 231*((b^14*c^4 - 28*a*b^12*c^5 + 33
6*a^2*b^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672
*a^6*b^2*c^10 - 16384*a^7*c^11)*d^26)^(1/4)*(c*x^2 + b*x + a)*log(-1331*(b^10*c^
3 - 20*a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a
^5*c^8)*sqrt(2*c*d*x + b*d)*d^19 + 1331*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^1
0*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*
c^10 - 16384*a^7*c^11)*d^26)^(3/4)) - 231*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b
^10*c^6 - 2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^
2*c^10 - 16384*a^7*c^11)*d^26)^(1/4)*(c*x^2 + b*x + a)*log(-1331*(b^10*c^3 - 20*
a*b^8*c^4 + 160*a^2*b^6*c^5 - 640*a^3*b^4*c^6 + 1280*a^4*b^2*c^7 - 1024*a^5*c^8)
*sqrt(2*c*d*x + b*d)*d^19 - 1331*((b^14*c^4 - 28*a*b^12*c^5 + 336*a^2*b^10*c^6 -
2240*a^3*b^8*c^7 + 8960*a^4*b^6*c^8 - 21504*a^5*b^4*c^9 + 28672*a^6*b^2*c^10 -
16384*a^7*c^11)*d^26)^(3/4)) + (384*c^5*d^6*x^5 + 960*b*c^4*d^6*x^4 + 32*(41*b^2
*c^3 - 44*a*c^4)*d^6*x^3 + 48*(21*b^3*c^2 - 44*a*b*c^3)*d^6*x^2 + 2*(115*b^4*c +
88*a*b^2*c^2 - 1232*a^2*c^3)*d^6*x - (21*b^5 - 440*a*b^3*c + 1232*a^2*b*c^2)*d^
6)*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)**(13/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265241, size = 767, normalized size = 4.24 \[ \frac{32}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c d^{5} - \frac{128}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{2} d^{5} + \frac{16}{7} \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c d^{3} + \frac{11}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{11}{2} \, \sqrt{2}{\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - 11 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - 11 \,{\left (\sqrt{2} b^{2} c d^{5} - 4 \, \sqrt{2} a c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) + \frac{4 \,{\left ({\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c d^{7} - 8 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} c^{2} d^{7} + 16 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} c^{3} d^{7}\right )}}{b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(13/2)/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

32/3*(2*c*d*x + b*d)^(3/2)*b^2*c*d^5 - 128/3*(2*c*d*x + b*d)^(3/2)*a*c^2*d^5 + 1
6/7*(2*c*d*x + b*d)^(7/2)*c*d^3 + 11/2*sqrt(2)*(b^2*c*d^5 - 4*a*c^2*d^5)*(-b^2*d
^2 + 4*a*c*d^2)^(3/4)*ln(2*c*d*x + b*d + sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*sq
rt(2*c*d*x + b*d) + sqrt(-b^2*d^2 + 4*a*c*d^2)) - 11/2*sqrt(2)*(b^2*c*d^5 - 4*a*
c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)) - 11*(sqrt(2)*b^
2*c*d^5 - 4*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*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)) - 11*(sqrt(2)*b^2*c*d^5 - 4*sqrt(2)*a*c^2*d^5)*(-b^2*d^2 + 4*a*c*d^2
)^(3/4)*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)) + 4*((2*c*d*x + b*d)^(3/2)*b^4*c*d^7 -
8*(2*c*d*x + b*d)^(3/2)*a*b^2*c^2*d^7 + 16*(2*c*d*x + b*d)^(3/2)*a^2*c^3*d^7)/(b
^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)