3.1298 \(\int \frac{(b d+2 c d x)^{13/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=193 \[ 77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{154}{3} c^2 d^5 (b d+2 c d x)^{3/2} \]

[Out]

(154*c^2*d^5*(b*d + 2*c*d*x)^(3/2))/3 - (d*(b*d + 2*c*d*x)^(11/2))/(2*(a + b*x +
c*x^2)^2) - (11*c*d^3*(b*d + 2*c*d*x)^(7/2))/(2*(a + b*x + c*x^2)) + 77*c^2*(b^
2 - 4*a*c)^(3/4)*d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d
])] - 77*c^2*(b^2 - 4*a*c)^(3/4)*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 = 0.404011, antiderivative size = 193, 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 \[ 77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-77 c^2 d^{13/2} \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-\frac{11 c d^3 (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{154}{3} c^2 d^5 (b d+2 c d x)^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^3,x]

[Out]

(154*c^2*d^5*(b*d + 2*c*d*x)^(3/2))/3 - (d*(b*d + 2*c*d*x)^(11/2))/(2*(a + b*x +
c*x^2)^2) - (11*c*d^3*(b*d + 2*c*d*x)^(7/2))/(2*(a + b*x + c*x^2)) + 77*c^2*(b^
2 - 4*a*c)^(3/4)*d^(13/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d
])] - 77*c^2*(b^2 - 4*a*c)^(3/4)*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 = 96.2499, size = 196, normalized size = 1.02 \[ 77 c^{2} d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 77 c^{2} d^{\frac{13}{2}} \left (- 4 a c + b^{2}\right )^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + \frac{154 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} - \frac{11 c d^{3} \left (b d + 2 c d x\right )^{\frac{7}{2}}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{11}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

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)**3,x)

[Out]

77*c**2*d**(13/2)*(-4*a*c + b**2)**(3/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a
*c + b**2)**(1/4))) - 77*c**2*d**(13/2)*(-4*a*c + b**2)**(3/4)*atanh(sqrt(b*d +
2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 154*c**2*d**5*(b*d + 2*c*d*x)**(3/2
)/3 - 11*c*d**3*(b*d + 2*c*d*x)**(7/2)/(2*(a + b*x + c*x**2)) - d*(b*d + 2*c*d*x
)**(11/2)/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 1.30729, size = 210, normalized size = 1.09 \[ (d (b+2 c x))^{13/2} \left (\frac{4 c^2 \left (77 a^2+121 a c x^2+32 c^2 x^4\right )+b^2 c \left (71 c x^2-33 a\right )+4 b c^2 x \left (121 a+64 c x^2\right )-3 b^4-57 b^3 c x}{6 (b+2 c x)^5 (a+x (b+c x))^2}+\frac{77 c^2 \left (b^2-4 a c\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}-\frac{77 c^2 \left (b^2-4 a c\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{13/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(13/2)/(a + b*x + c*x^2)^3,x]

[Out]

(d*(b + 2*c*x))^(13/2)*((-3*b^4 - 57*b^3*c*x + 4*b*c^2*x*(121*a + 64*c*x^2) + b^
2*c*(-33*a + 71*c*x^2) + 4*c^2*(77*a^2 + 121*a*c*x^2 + 32*c^2*x^4))/(6*(b + 2*c*
x)^5*(a + x*(b + c*x))^2) + (77*c^2*(b^2 - 4*a*c)^(3/4)*ArcTan[Sqrt[b + 2*c*x]/(
b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(13/2) - (77*c^2*(b^2 - 4*a*c)^(3/4)*ArcTanh[Sq
rt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(13/2))

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Maple [B]  time = 0.024, size = 857, normalized size = 4.4 \[ \text{result too large to display} \]

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)^3,x)

[Out]

64/3*c^2*d^5*(2*c*d*x+b*d)^(3/2)+152*c^3*d^7/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^
2)^2*(2*c*d*x+b*d)^(7/2)*a-38*c^2*d^7/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2
*c*d*x+b*d)^(7/2)*b^2+480*c^4*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d
*x+b*d)^(3/2)*a^2-240*c^3*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d*x+b
*d)^(3/2)*a*b^2+30*c^2*d^9/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2*(2*c*d*x+b*d)
^(3/2)*b^4-77*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a*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)))-154*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*a*arctan(2^(1/2)/(4*a*c*
d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)+154*c^3*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*
2^(1/2)*a*arctan(-2^(1/2)/(4*a*c*d^2-b^2*d^2)^(1/4)*(2*c*d*x+b*d)^(1/2)+1)+77/4*
c^2*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*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)))+77
/2*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*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)-77/2*c^2*d^7/(4*a*c*d^2-b^2*d^2)^(1/4)*2^(1/2)*
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)

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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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243336, size = 1173, normalized size = 6.08 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

-1/6*(924*((b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)^(1/4)*
(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*arctan(((b^6*c^8 - 12*
a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)^(3/4)/((b^4*c^6 - 8*a*b^2*c^7 +
16*a^2*c^8)*sqrt(2*c*d*x + b*d)*d^19 + sqrt(2*(b^8*c^13 - 16*a*b^6*c^14 + 96*a^
2*b^4*c^15 - 256*a^3*b^2*c^16 + 256*a^4*c^17)*d^39*x + (b^9*c^12 - 16*a*b^7*c^13
+ 96*a^2*b^5*c^14 - 256*a^3*b^3*c^15 + 256*a^4*b*c^16)*d^39 + sqrt((b^6*c^8 - 1
2*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)*(b^6*c^8 - 12*a*b^4*c^9 + 48*
a^2*b^2*c^10 - 64*a^3*c^11)*d^26))) + 231*((b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*
c^10 - 64*a^3*c^11)*d^26)^(1/4)*(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x
^2 + a^2)*log(456533*(b^4*c^6 - 8*a*b^2*c^7 + 16*a^2*c^8)*sqrt(2*c*d*x + b*d)*d^
19 + 456533*((b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)^(3/4
)) - 231*((b^6*c^8 - 12*a*b^4*c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)^(1/4)*(
c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*log(456533*(b^4*c^6 - 8
*a*b^2*c^7 + 16*a^2*c^8)*sqrt(2*c*d*x + b*d)*d^19 - 456533*((b^6*c^8 - 12*a*b^4*
c^9 + 48*a^2*b^2*c^10 - 64*a^3*c^11)*d^26)^(3/4)) - (256*c^5*d^6*x^5 + 640*b*c^4
*d^6*x^4 + 2*(199*b^2*c^3 + 484*a*c^4)*d^6*x^3 - (43*b^3*c^2 - 1452*a*b*c^3)*d^6
*x^2 - (63*b^4*c - 418*a*b^2*c^2 - 616*a^2*c^3)*d^6*x - (3*b^5 + 33*a*b^3*c - 30
8*a^2*b*c^2)*d^6)*sqrt(2*c*d*x + b*d))/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2
*a*c)*x^2 + a^2)

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.269915, size = 703, normalized size = 3.64 \[ -\frac{77}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{5} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) - \frac{77}{2} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{5} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) + \frac{77}{4} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{5}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{77}{4} \, \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{5}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{64}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{2} d^{5} + \frac{2 \,{\left (15 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{4} c^{2} d^{9} - 120 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a b^{2} c^{3} d^{9} + 240 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a^{2} c^{4} d^{9} - 19 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} b^{2} c^{2} d^{7} + 76 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} a c^{3} d^{7}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

-77/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^5*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
)) - 77/2*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^5*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)) + 77/4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^5*ln(2*c*d*x + b*d + s
qrt(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)) - 77/4*sqrt(2)*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^5*ln(2*c*d*x + b*d - sqr
t(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)) + 64/3*(2*c*d*x + b*d)^(3/2)*c^2*d^5 + 2*(15*(2*c*d*x + b*d)^(3/2)*b^4*c^2*d
^9 - 120*(2*c*d*x + b*d)^(3/2)*a*b^2*c^3*d^9 + 240*(2*c*d*x + b*d)^(3/2)*a^2*c^4
*d^9 - 19*(2*c*d*x + b*d)^(7/2)*b^2*c^2*d^7 + 76*(2*c*d*x + b*d)^(7/2)*a*c^3*d^7
)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2