3.1275 \(\int \frac{(b d+2 c d x)^{9/2}}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=147 \[ 2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2} \]

[Out]

(4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2))/3 + (4*d*(b*d + 2*c*d*x)^(7/2))/7 +
2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])] - 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.351515, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ 2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-2 d^{9/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+\frac{4}{3} d^3 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}+\frac{4}{7} d (b d+2 c d x)^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]

[Out]

(4*(b^2 - 4*a*c)*d^3*(b*d + 2*c*d*x)^(3/2))/3 + (4*d*(b*d + 2*c*d*x)^(7/2))/7 +
2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sq
rt[d])] - 2*(b^2 - 4*a*c)^(7/4)*d^(9/2)*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 = 79.4087, size = 148, normalized size = 1.01 \[ 2 d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 2 d^{\frac{9}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + \frac{4 d^{3} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}}}{3} + \frac{4 d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a),x)

[Out]

2*d**(9/2)*(-4*a*c + b**2)**(7/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b*
*2)**(1/4))) - 2*d**(9/2)*(-4*a*c + b**2)**(7/4)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt
(d)*(-4*a*c + b**2)**(1/4))) + 4*d**3*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)/3 +
4*d*(b*d + 2*c*d*x)**(7/2)/7

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Mathematica [A]  time = 0.329688, size = 162, normalized size = 1.1 \[ \frac{2 d^4 \sqrt{d (b+2 c x)} \left (4 \sqrt [4]{b^2-4 a c} (b+2 c x)^{3/2} \left (2 c \left (3 c x^2-7 a\right )+5 b^2+6 b c x\right )+21 \left (b^2-4 a c\right )^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )-21 \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )\right )}{21 \sqrt [4]{b^2-4 a c} \sqrt{b+2 c x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2),x]

[Out]

(2*d^4*Sqrt[d*(b + 2*c*x)]*(4*(b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(3/2)*(5*b^2 + 6*b
*c*x + 2*c*(-7*a + 3*c*x^2)) + 21*(b^2 - 4*a*c)^2*ArcTan[Sqrt[b + 2*c*x]/(b^2 -
4*a*c)^(1/4)] - 21*(b^2 - 4*a*c)^2*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])
)/(21*(b^2 - 4*a*c)^(1/4)*Sqrt[b + 2*c*x])

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Maple [B]  time = 0.012, size = 922, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a),x)

[Out]

4/7*d*(2*c*d*x+b*d)^(7/2)-16/3*(2*c*d*x+b*d)^(3/2)*a*c*d^3+4/3*b^2*d^3*(2*c*d*x+
b*d)^(3/2)-16*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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*c^2+8*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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*c-d^
5/(4*a*c*d^2-b^2*d^2)^(1/4)*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+8*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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*c^2-4*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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*c+1/2*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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+16*d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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*c^2-8*d^5/(4*a*c*d^2-b^2*d^2
)^(1/4)*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*c+d^5/(4*a*c*d^2-b^2*d^2)^(1/4)*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)^(9/2)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242493, size = 1571, normalized size = 10.69 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a),x, algorithm="fricas")

[Out]

8/21*(12*c^3*d^4*x^3 + 18*b*c^2*d^4*x^2 + 4*(4*b^2*c - 7*a*c^2)*d^4*x + (5*b^3 -
14*a*b*c)*d^4)*sqrt(2*c*d*x + b*d) + 4*((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2
- 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 -
16384*a^7*c^7)*d^18)^(1/4)*arctan(-((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 224
0*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384
*a^7*c^7)*d^18)^(3/4)/((b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 +
1280*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 - sqrt(2*(b^20*c - 40*
a*b^18*c^2 + 720*a^2*b^16*c^3 - 7680*a^3*b^14*c^4 + 53760*a^4*b^12*c^5 - 258048*
a^5*b^10*c^6 + 860160*a^6*b^8*c^7 - 1966080*a^7*b^6*c^8 + 2949120*a^8*b^4*c^9 -
2621440*a^9*b^2*c^10 + 1048576*a^10*c^11)*d^27*x + (b^21 - 40*a*b^19*c + 720*a^2
*b^17*c^2 - 7680*a^3*b^15*c^3 + 53760*a^4*b^13*c^4 - 258048*a^5*b^11*c^5 + 86016
0*a^6*b^9*c^6 - 1966080*a^7*b^7*c^7 + 2949120*a^8*b^5*c^8 - 2621440*a^9*b^3*c^9
+ 1048576*a^10*b*c^10)*d^27 + sqrt((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240
*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*
a^7*c^7)*d^18)*(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*
a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18))) + (
(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 2
1504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(1/4)*log(-(b^10 - 2
0*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*c^4 - 1024*a^5*c^5)
*sqrt(2*c*d*x + b*d)*d^13 + ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b
^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^
7)*d^18)^(3/4)) - ((b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8
960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7)*d^18)^(
1/4)*log(-(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280*a^4*b^2*
c^4 - 1024*a^5*c^5)*sqrt(2*c*d*x + b*d)*d^13 - ((b^14 - 28*a*b^12*c + 336*a^2*b^
10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2
*c^6 - 16384*a^7*c^7)*d^18)^(3/4))

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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)**(9/2)/(c*x**2+b*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.258508, size = 612, normalized size = 4.16 \[ \frac{4}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} d^{3} - \frac{16}{3} \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c d^{3} + \frac{4}{7} \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} d + \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) - \frac{1}{2} \, \sqrt{2}{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) -{\left (\sqrt{2} b^{2} d^{3} - 4 \, \sqrt{2} a c d^{3}\right )}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

4/3*(2*c*d*x + b*d)^(3/2)*b^2*d^3 - 16/3*(2*c*d*x + b*d)^(3/2)*a*c*d^3 + 4/7*(2*
c*d*x + b*d)^(7/2)*d + 1/2*sqrt(2)*(b^2*d^3 - 4*a*c*d^3)*(-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)) - 1/2*sqrt(2)*(b^2*d^3 - 4*a*c*d^3)*(-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)) - (sqrt(2)*b^2*d^3 - 4*sqrt(2)*a*c*d^
3)*(-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)) - (sqrt(2)*b^2*d
^3 - 4*sqrt(2)*a*c*d^3)*(-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))