3.1296 \(\int \frac{(b d+2 c d x)^{17/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=222 \[ 165 c^2 d^{17/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-165 c^2 d^{17/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+110 c^2 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}-\frac{15 c d^3 (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{330}{7} c^2 d^5 (b d+2 c d x)^{7/2} \]

[Out]

110*c^2*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(3/2) + (330*c^2*d^5*(b*d + 2*c*d*x)^(
7/2))/7 - (d*(b*d + 2*c*d*x)^(15/2))/(2*(a + b*x + c*x^2)^2) - (15*c*d^3*(b*d +
2*c*d*x)^(11/2))/(2*(a + b*x + c*x^2)) + 165*c^2*(b^2 - 4*a*c)^(7/4)*d^(17/2)*Ar
cTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 165*c^2*(b^2 - 4*a*c)^
(7/4)*d^(17/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.510932, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ 165 c^2 d^{17/2} \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-165 c^2 d^{17/2} \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+110 c^2 d^7 \left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}-\frac{15 c d^3 (b d+2 c d x)^{11/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{15/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{330}{7} c^2 d^5 (b d+2 c d x)^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x]

[Out]

110*c^2*(b^2 - 4*a*c)*d^7*(b*d + 2*c*d*x)^(3/2) + (330*c^2*d^5*(b*d + 2*c*d*x)^(
7/2))/7 - (d*(b*d + 2*c*d*x)^(15/2))/(2*(a + b*x + c*x^2)^2) - (15*c*d^3*(b*d +
2*c*d*x)^(11/2))/(2*(a + b*x + c*x^2)) + 165*c^2*(b^2 - 4*a*c)^(7/4)*d^(17/2)*Ar
cTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 165*c^2*(b^2 - 4*a*c)^
(7/4)*d^(17/2)*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 = 116.599, size = 226, normalized size = 1.02 \[ 165 c^{2} d^{\frac{17}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 165 c^{2} d^{\frac{17}{2}} \left (- 4 a c + b^{2}\right )^{\frac{7}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 110 c^{2} d^{7} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} + \frac{330 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac{7}{2}}}{7} - \frac{15 c d^{3} \left (b d + 2 c d x\right )^{\frac{11}{2}}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{15}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(17/2)/(c*x**2+b*x+a)**3,x)

[Out]

165*c**2*d**(17/2)*(-4*a*c + b**2)**(7/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*
a*c + b**2)**(1/4))) - 165*c**2*d**(17/2)*(-4*a*c + b**2)**(7/4)*atanh(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 110*c**2*d**7*(-4*a*c + b**2)*(b*
d + 2*c*d*x)**(3/2) + 330*c**2*d**5*(b*d + 2*c*d*x)**(7/2)/7 - 15*c*d**3*(b*d +
2*c*d*x)**(11/2)/(2*(a + b*x + c*x**2)) - d*(b*d + 2*c*d*x)**(15/2)/(2*(a + b*x
+ c*x**2)**2)

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Mathematica [A]  time = 1.25676, size = 281, normalized size = 1.27 \[ (d (b+2 c x))^{17/2} \left (\frac{40 b^2 c^2 \left (55 a^2+25 a c x^2+64 c^2 x^4\right )+16 b c^3 x \left (-605 a^2-320 a c x^2+96 c^2 x^4\right )-16 c^3 \left (385 a^3+605 a^2 c x^2+160 a c^2 x^4-32 c^3 x^6\right )+5 b^4 c \left (167 c x^2-21 a\right )+40 b^3 c^2 x \left (89 a+64 c x^2\right )-7 b^6-189 b^5 c x}{14 (b+2 c x)^7 (a+x (b+c x))^2}+\frac{165 c^2 \left (b^2-4 a c\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{17/2}}-\frac{165 c^2 \left (b^2-4 a c\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{17/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(17/2)/(a + b*x + c*x^2)^3,x]

[Out]

(d*(b + 2*c*x))^(17/2)*((-7*b^6 - 189*b^5*c*x + 40*b^3*c^2*x*(89*a + 64*c*x^2) +
5*b^4*c*(-21*a + 167*c*x^2) + 40*b^2*c^2*(55*a^2 + 25*a*c*x^2 + 64*c^2*x^4) + 1
6*b*c^3*x*(-605*a^2 - 320*a*c*x^2 + 96*c^2*x^4) - 16*c^3*(385*a^3 + 605*a^2*c*x^
2 + 160*a*c^2*x^4 - 32*c^3*x^6))/(14*(b + 2*c*x)^7*(a + x*(b + c*x))^2) + (165*c
^2*(b^2 - 4*a*c)^(7/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^
(17/2) - (165*c^2*(b^2 - 4*a*c)^(7/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4
)])/(b + 2*c*x)^(17/2))

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Maple [B]  time = 0.027, size = 1310, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(17/2)/(c*x^2+b*x+a)^3,x)

[Out]

64/7*c^2*d^5*(2*c*d*x+b*d)^(7/2)-256*c^3*d^7*(2*c*d*x+b*d)^(3/2)*a+64*c^2*d^7*b^
2*(2*c*d*x+b*d)^(3/2)-864*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)^(7/2)*a^2+432*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)^(7/2)*a*b^2-54*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)
^(7/2)*b^4-2944*c^5*d^11/(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^3+2208*c^4*d^11/(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*b^2-552*c^3*d^11/(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^4+46*c^2*d^11/(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^6+660*c^4*d^9/(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)))+1320*c^4*d^9/(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)-1320*c^4*d^9/(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)-3
30*c^3*d^9/(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)
))-660*c^3*d^9/(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)+660*c^3*d^9/(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)+165/
4*c^2*d^9/(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)))+
165/2*c^2*d^9/(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)-165/2*c^2*d^9/(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)^(17/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25331, size = 2084, normalized size = 9.39 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(17/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

1/14*(4620*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 +
8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^
34)^(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^
14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^
12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^34)^(3/4)/((b^1
0*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1
024*a^5*c^11)*sqrt(2*c*d*x + b*d)*d^25 - sqrt(2*(b^20*c^13 - 40*a*b^18*c^14 + 72
0*a^2*b^16*c^15 - 7680*a^3*b^14*c^16 + 53760*a^4*b^12*c^17 - 258048*a^5*b^10*c^1
8 + 860160*a^6*b^8*c^19 - 1966080*a^7*b^6*c^20 + 2949120*a^8*b^4*c^21 - 2621440*
a^9*b^2*c^22 + 1048576*a^10*c^23)*d^51*x + (b^21*c^12 - 40*a*b^19*c^13 + 720*a^2
*b^17*c^14 - 7680*a^3*b^15*c^15 + 53760*a^4*b^13*c^16 - 258048*a^5*b^11*c^17 + 8
60160*a^6*b^9*c^18 - 1966080*a^7*b^7*c^19 + 2949120*a^8*b^5*c^20 - 2621440*a^9*b
^3*c^21 + 1048576*a^10*b*c^22)*d^51 + sqrt((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b
^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^
6*b^2*c^14 - 16384*a^7*c^15)*d^34)*(b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10
- 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^
14 - 16384*a^7*c^15)*d^34))) + 1155*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^
10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*
c^14 - 16384*a^7*c^15)*d^34)^(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(-4492125*(b^10*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b
^4*c^9 + 1280*a^4*b^2*c^10 - 1024*a^5*c^11)*sqrt(2*c*d*x + b*d)*d^25 + 4492125*(
(b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6
*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^34)^(3/4)) -
1155*((b^14*c^8 - 28*a*b^12*c^9 + 336*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*
a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 28672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^34)^(
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(-4492125*(b^1
0*c^6 - 20*a*b^8*c^7 + 160*a^2*b^6*c^8 - 640*a^3*b^4*c^9 + 1280*a^4*b^2*c^10 - 1
024*a^5*c^11)*sqrt(2*c*d*x + b*d)*d^25 - 4492125*((b^14*c^8 - 28*a*b^12*c^9 + 33
6*a^2*b^10*c^10 - 2240*a^3*b^8*c^11 + 8960*a^4*b^6*c^12 - 21504*a^5*b^4*c^13 + 2
8672*a^6*b^2*c^14 - 16384*a^7*c^15)*d^34)^(3/4)) + (1024*c^7*d^8*x^7 + 3584*b*c^
6*d^8*x^6 + 512*(13*b^2*c^5 - 10*a*c^6)*d^8*x^5 + 2560*(3*b^3*c^4 - 5*a*b*c^5)*d
^8*x^4 + 10*(423*b^4*c^3 - 312*a*b^2*c^4 - 1936*a^2*c^5)*d^8*x^3 + (457*b^5*c^2
+ 8120*a*b^3*c^3 - 29040*a^2*b*c^4)*d^8*x^2 - (203*b^6*c - 3350*a*b^4*c^2 + 5280
*a^2*b^2*c^3 + 12320*a^3*c^4)*d^8*x - (7*b^7 + 105*a*b^5*c - 2200*a^2*b^3*c^2 +
6160*a^3*b*c^3)*d^8)*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)**(17/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286829, size = 903, normalized size = 4.07 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(17/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

64*(2*c*d*x + b*d)^(3/2)*b^2*c^2*d^7 - 256*(2*c*d*x + b*d)^(3/2)*a*c^3*d^7 + 64/
7*(2*c*d*x + b*d)^(7/2)*c^2*d^5 - 165/2*sqrt(2)*(b^2*c^2*d^7 - 4*a*c^3*d^7)*(-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)) - 165/2*sqrt(2)*(b^2*c^
2*d^7 - 4*a*c^3*d^7)*(-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
)) + 165/4*sqrt(2)*(b^2*c^2*d^7 - 4*a*c^3*d^7)*(-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)) - 165/4*sqrt(2)*(b^2*c^2*d^7 - 4*a*c^3*d^7)*(-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)) + 2*(23*(2*c*d*x + b*d)^(3/2)*b^6*c^2*d
^11 - 276*(2*c*d*x + b*d)^(3/2)*a*b^4*c^3*d^11 + 1104*(2*c*d*x + b*d)^(3/2)*a^2*
b^2*c^4*d^11 - 1472*(2*c*d*x + b*d)^(3/2)*a^3*c^5*d^11 - 27*(2*c*d*x + b*d)^(7/2
)*b^4*c^2*d^9 + 216*(2*c*d*x + b*d)^(7/2)*a*b^2*c^3*d^9 - 432*(2*c*d*x + b*d)^(7
/2)*a^2*c^4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2