3.1274 \(\int \frac{(b d+2 c d x)^{11/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=175 \[ -2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d^5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}+\frac{4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}+\frac{4}{9} d (b d+2 c d x)^{9/2} \]

[Out]

4*(b^2 - 4*a*c)^2*d^5*Sqrt[b*d + 2*c*d*x] + (4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)
^(5/2))/5 + (4*d*(b*d + 2*c*d*x)^(9/2))/9 - 2*(b^2 - 4*a*c)^(9/4)*d^(11/2)*ArcTa
n[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(9/4)*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 = 0.487556, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{11/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+4 d^5 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}+\frac{4}{5} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}+\frac{4}{9} d (b d+2 c d x)^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

4*(b^2 - 4*a*c)^2*d^5*Sqrt[b*d + 2*c*d*x] + (4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)
^(5/2))/5 + (4*d*(b*d + 2*c*d*x)^(9/2))/9 - 2*(b^2 - 4*a*c)^(9/4)*d^(11/2)*ArcTa
n[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 2*(b^2 - 4*a*c)^(9/4)*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 = 95.7899, size = 177, normalized size = 1.01 \[ - 2 d^{\frac{11}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 2 d^{\frac{11}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 4 d^{5} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{9}{2}}}{9} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.431725, size = 190, normalized size = 1.09 \[ \frac{2 d^5 \sqrt{d (b+2 c x)} \left (2 \sqrt{b+2 c x} \left (16 c^2 \left (45 a^2-9 a c x^2+5 c^2 x^4\right )+12 b^2 c \left (13 c x^2-33 a\right )+16 b c^2 x \left (10 c x^2-9 a\right )+59 b^4+76 b^3 c x\right )-45 \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-45 \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{45 \sqrt{b+2 c x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*d^5*Sqrt[d*(b + 2*c*x)]*(2*Sqrt[b + 2*c*x]*(59*b^4 + 76*b^3*c*x + 16*b*c^2*x*
(-9*a + 10*c*x^2) + 12*b^2*c*(-33*a + 13*c*x^2) + 16*c^2*(45*a^2 - 9*a*c*x^2 + 5
*c^2*x^4)) - 45*(b^2 - 4*a*c)^(9/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]
- 45*(b^2 - 4*a*c)^(9/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)]))/(45*Sqrt
[b + 2*c*x])

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Maple [B]  time = 0.027, size = 1287, normalized size = 7.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/9*d*(2*c*d*x+b*d)^(9/2)-16/5*(2*c*d*x+b*d)^(5/2)*a*c*d^3+4/5*b^2*(2*c*d*x+b*d)
^(5/2)*d^3+64*a^2*c^2*d^5*(2*c*d*x+b*d)^(1/2)-32*a*b^2*c*d^5*(2*c*d*x+b*d)^(1/2)
+4*b^4*d^5*(2*c*d*x+b*d)^(1/2)+64*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^3*c^3-48*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*b^2*c^2+12*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^4*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^6-32*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^3
*c^3+24*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*b^2*c^2-6*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^4*c+1/2*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
)))*b^6-64*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^3*c^3+48*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*b^2*c^2-12
*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^4*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^6

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244869, size = 1401, normalized size = 8.01 \[ \text{result too large to display} \]

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),x, algorithm="fricas")

[Out]

4/45*(80*c^4*d^5*x^4 + 160*b*c^3*d^5*x^3 + 12*(13*b^2*c^2 - 12*a*c^3)*d^5*x^2 +
4*(19*b^3*c - 36*a*b*c^2)*d^5*x + (59*b^4 - 396*a*b^2*c + 720*a^2*c^2)*d^5)*sqrt
(2*c*d*x + b*d) + 4*((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3
+ 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*
c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*arctan(((b^18 - 36*a*b^16
*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*
c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*
c^9)*d^22)^(1/4)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^5 + sqrt(
2*(b^8*c - 16*a*b^6*c^2 + 96*a^2*b^4*c^3 - 256*a^3*b^2*c^4 + 256*a^4*c^5)*d^11*x
+ (b^9 - 16*a*b^7*c + 96*a^2*b^5*c^2 - 256*a^3*b^3*c^3 + 256*a^4*b*c^4)*d^11 +
sqrt((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10
*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8
*b^2*c^8 - 262144*a^9*c^9)*d^22)))) - ((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 -
5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6
- 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*log((b^
4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(2*c*d*x + b*d)*d^5 + ((b^18 - 36*a*b^16*c + 576
*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 34
4064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^2
2)^(1/4)) + ((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b^12*c^3 + 32256*
a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*a^7*b^4*c^7 + 58
9824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4)*log((b^4 - 8*a*b^2*c + 16*a^2*c^2
)*sqrt(2*c*d*x + b*d)*d^5 - ((b^18 - 36*a*b^16*c + 576*a^2*b^14*c^2 - 5376*a^3*b
^12*c^3 + 32256*a^4*b^10*c^4 - 129024*a^5*b^8*c^5 + 344064*a^6*b^6*c^6 - 589824*
a^7*b^4*c^7 + 589824*a^8*b^2*c^8 - 262144*a^9*c^9)*d^22)^(1/4))

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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),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263549, size = 779, normalized size = 4.45 \[ 4 \, \sqrt{2 \, c d x + b d} b^{4} d^{5} - 32 \, \sqrt{2 \, c d x + b d} a b^{2} c d^{5} + 64 \, \sqrt{2 \, c d x + b d} a^{2} c^{2} d^{5} + \frac{4}{5} \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} b^{2} d^{3} - \frac{16}{5} \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} a c d^{3} + \frac{4}{9} \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} d - \frac{1}{2} \, \sqrt{2}{\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) + \frac{1}{2} \, \sqrt{2}{\left (b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) -{\left (\sqrt{2} b^{4} d^{5} - 8 \, \sqrt{2} a b^{2} c d^{5} + 16 \, \sqrt{2} a^{2} c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (\sqrt{2} b^{4} d^{5} - 8 \, \sqrt{2} a b^{2} c d^{5} + 16 \, \sqrt{2} a^{2} c^{2} d^{5}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) \]

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),x, algorithm="giac")

[Out]

4*sqrt(2*c*d*x + b*d)*b^4*d^5 - 32*sqrt(2*c*d*x + b*d)*a*b^2*c*d^5 + 64*sqrt(2*c
*d*x + b*d)*a^2*c^2*d^5 + 4/5*(2*c*d*x + b*d)^(5/2)*b^2*d^3 - 16/5*(2*c*d*x + b*
d)^(5/2)*a*c*d^3 + 4/9*(2*c*d*x + b*d)^(9/2)*d - 1/2*sqrt(2)*(b^4*d^5 - 8*a*b^2*
c*d^5 + 16*a^2*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)) +
1/2*sqrt(2)*(b^4*d^5 - 8*a*b^2*c*d^5 + 16*a^2*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)) - (sqrt(2)*b^4*d^5 - 8*sqrt(2)*a*b^2*c*d^5 + 16*s
qrt(2)*a^2*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))
- (sqrt(2)*b^4*d^5 - 8*sqrt(2)*a*b^2*c*d^5 + 16*sqrt(2)*a^2*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))