3.1284 \(\int \frac{(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=210 \[ -26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+52 c d^7 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}+\frac{52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-\frac{d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac{52}{9} c d^3 (b d+2 c d x)^{9/2} \]

[Out]

52*c*(b^2 - 4*a*c)^2*d^7*Sqrt[b*d + 2*c*d*x] + (52*c*(b^2 - 4*a*c)*d^5*(b*d + 2*
c*d*x)^(5/2))/5 + (52*c*d^3*(b*d + 2*c*d*x)^(9/2))/9 - (d*(b*d + 2*c*d*x)^(13/2)
)/(a + b*x + c*x^2) - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcTan[Sqrt[d*(b + 2*c*x
)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcTanh[Sq
rt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.479597, antiderivative size = 210, 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 \[ -26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-26 c d^{15/2} \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+52 c d^7 \left (b^2-4 a c\right )^2 \sqrt{b d+2 c d x}+\frac{52}{5} c d^5 \left (b^2-4 a c\right ) (b d+2 c d x)^{5/2}-\frac{d (b d+2 c d x)^{13/2}}{a+b x+c x^2}+\frac{52}{9} c d^3 (b d+2 c d x)^{9/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x]

[Out]

52*c*(b^2 - 4*a*c)^2*d^7*Sqrt[b*d + 2*c*d*x] + (52*c*(b^2 - 4*a*c)*d^5*(b*d + 2*
c*d*x)^(5/2))/5 + (52*c*d^3*(b*d + 2*c*d*x)^(9/2))/9 - (d*(b*d + 2*c*d*x)^(13/2)
)/(a + b*x + c*x^2) - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcTan[Sqrt[d*(b + 2*c*x
)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 26*c*(b^2 - 4*a*c)^(9/4)*d^(15/2)*ArcTanh[Sq
rt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi in Sympy [A]  time = 115.442, size = 212, normalized size = 1.01 \[ - 26 c d^{\frac{15}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 26 c d^{\frac{15}{2}} \left (- 4 a c + b^{2}\right )^{\frac{9}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 52 c d^{7} \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} + \frac{52 c d^{5} \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} + \frac{52 c d^{3} \left (b d + 2 c d x\right )^{\frac{9}{2}}}{9} - \frac{d \left (b d + 2 c d x\right )^{\frac{13}{2}}}{a + b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**2,x)

[Out]

-26*c*d**(15/2)*(-4*a*c + b**2)**(9/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4))) - 26*c*d**(15/2)*(-4*a*c + b**2)**(9/4)*atanh(sqrt(b*d + 2*c*d
*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 52*c*d**7*(-4*a*c + b**2)**2*sqrt(b*d +
2*c*d*x) + 52*c*d**5*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)/5 + 52*c*d**3*(b*d +
2*c*d*x)**(9/2)/9 - d*(b*d + 2*c*d*x)**(13/2)/(a + b*x + c*x**2)

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Mathematica [A]  time = 1.14529, size = 224, normalized size = 1.07 \[ (d (b+2 c x))^{15/2} \left (\frac{32 c \left (1080 a^2 c^2-576 a b^2 c+79 b^4\right )-1536 c^3 x^2 \left (3 a c-2 b^2\right )+256 b c^2 x \left (7 b^2-18 a c\right )-\frac{45 \left (b^2-4 a c\right )^3}{a+x (b+c x)}+2560 b c^4 x^3+1280 c^5 x^4}{45 (b+2 c x)^7}-\frac{26 c \left (b^2-4 a c\right )^{9/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}-\frac{26 c \left (b^2-4 a c\right )^{9/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^2,x]

[Out]

(d*(b + 2*c*x))^(15/2)*((32*c*(79*b^4 - 576*a*b^2*c + 1080*a^2*c^2) + 256*b*c^2*
(7*b^2 - 18*a*c)*x - 1536*c^3*(-2*b^2 + 3*a*c)*x^2 + 2560*b*c^4*x^3 + 1280*c^5*x
^4 - (45*(b^2 - 4*a*c)^3)/(a + x*(b + c*x)))/(45*(b + 2*c*x)^7) - (26*c*(b^2 - 4
*a*c)^(9/4)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) - (2
6*c*(b^2 - 4*a*c)^(9/4)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x
)^(15/2))

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Maple [B]  time = 0.023, size = 1512, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(15/2)/(c*x^2+b*x+a)^2,x)

[Out]

16/9*c*d^3*(2*c*d*x+b*d)^(9/2)-128/5*c^2*d^5*(2*c*d*x+b*d)^(5/2)*a+32/5*c*d^5*b^
2*(2*c*d*x+b*d)^(5/2)+768*c^3*d^7*a^2*(2*c*d*x+b*d)^(1/2)-384*c^2*d^7*a*b^2*(2*c
*d*x+b*d)^(1/2)+48*c*d^7*b^4*(2*c*d*x+b*d)^(1/2)+256*c^4*d^9*(2*c*d*x+b*d)^(1/2)
/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*a^3-192*c^3*d^9*(2*c*d*x+b*d)^(1/2)/(4*c^
2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*a^2*b^2+48*c^2*d^9*(2*c*d*x+b*d)^(1/2)/(4*c^2*d
^2*x^2+4*b*c*d^2*x+4*a*c*d^2)*a*b^4-4*c*d^9*(2*c*d*x+b*d)^(1/2)/(4*c^2*d^2*x^2+4
*b*c*d^2*x+4*a*c*d^2)*b^6-832*c^4*d^9/(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+624*c^3*d^9/(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^2-156*c^2*d^9/(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+13*c*d^9/(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+832*c^4*d^9/(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^3-624*c^3*d^9/(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+1
56*c^2*d^9/(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^4-13*c*d^9/(4*a*c*d^2-b^2*d^2)^(3/4)*2^(1/2)*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)*b^6-416*c^4*d^9/
(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+312*c^3*d^
9/(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^2-78*c
^2*d^9/(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+1
3/2*c*d^9/(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

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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)^(15/2)/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247637, size = 1659, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

1/45*(2340*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 3
2256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^
11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)*(c*x^2 + b*x + a)*arctan
(((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b
^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 58982
4*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)/((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3
)*sqrt(2*c*d*x + b*d)*d^7 + sqrt(2*(b^8*c^3 - 16*a*b^6*c^4 + 96*a^2*b^4*c^5 - 25
6*a^3*b^2*c^6 + 256*a^4*c^7)*d^15*x + (b^9*c^2 - 16*a*b^7*c^3 + 96*a^2*b^5*c^4 -
256*a^3*b^3*c^5 + 256*a^4*b*c^6)*d^15 + sqrt((b^18*c^4 - 36*a*b^16*c^5 + 576*a^
2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 34406
4*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^
30)))) - 585*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 +
32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*
c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)*(c*x^2 + b*x + a)*log(
13*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 13*((b^18*c^4 -
36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 1290
24*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12
- 262144*a^9*c^13)*d^30)^(1/4)) + 585*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14
*c^6 - 5376*a^3*b^12*c^7 + 32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*
b^6*c^10 - 589824*a^7*b^4*c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1
/4)*(c*x^2 + b*x + a)*log(13*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*sqrt(2*c*d*x + b
*d)*d^7 - 13*((b^18*c^4 - 36*a*b^16*c^5 + 576*a^2*b^14*c^6 - 5376*a^3*b^12*c^7 +
32256*a^4*b^10*c^8 - 129024*a^5*b^8*c^9 + 344064*a^6*b^6*c^10 - 589824*a^7*b^4*
c^11 + 589824*a^8*b^2*c^12 - 262144*a^9*c^13)*d^30)^(1/4)) + (1280*c^6*d^7*x^6 +
3840*b*c^5*d^7*x^5 + 256*(22*b^2*c^4 - 13*a*c^5)*d^7*x^4 + 256*(19*b^3*c^3 - 26
*a*b*c^4)*d^7*x^3 + 96*(45*b^4*c^2 - 208*a*b^2*c^3 + 312*a^2*c^4)*d^7*x^2 + 32*(
79*b^5*c - 520*a*b^3*c^2 + 936*a^2*b*c^3)*d^7*x - (45*b^6 - 3068*a*b^4*c + 20592
*a^2*b^2*c^2 - 37440*a^3*c^3)*d^7)*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)**(15/2)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.264586, size = 972, normalized size = 4.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(15/2)/(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

48*sqrt(2*c*d*x + b*d)*b^4*c*d^7 - 384*sqrt(2*c*d*x + b*d)*a*b^2*c^2*d^7 + 768*s
qrt(2*c*d*x + b*d)*a^2*c^3*d^7 + 32/5*(2*c*d*x + b*d)^(5/2)*b^2*c*d^5 - 128/5*(2
*c*d*x + b*d)^(5/2)*a*c^2*d^5 + 16/9*(2*c*d*x + b*d)^(9/2)*c*d^3 - 13/2*sqrt(2)*
(b^4*c*d^7 - 8*a*b^2*c^2*d^7 + 16*a^2*c^3*d^7)*(-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)) + 13/2*sqrt(2)*(b^4*c*d^7 - 8*a*b^2*c^2*d^7 + 16*a^2*c^3*d
^7)*(-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)) - 13*(sqrt(2)*b^4*c*d
^7 - 8*sqrt(2)*a*b^2*c^2*d^7 + 16*sqrt(2)*a^2*c^3*d^7)*(-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)) - 13*(sqrt(2)*b^4*c*d^7 - 8*sqrt(2)*a*b^2*c^
2*d^7 + 16*sqrt(2)*a^2*c^3*d^7)*(-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)) + 4*(sqrt(2*c*d*x + b*d)*b^6*c*d^9 - 12*sqrt(2*c*d*x + b*d)*a*b^4*
c^2*d^9 + 48*sqrt(2*c*d*x + b*d)*a^2*b^2*c^3*d^9 - 64*sqrt(2*c*d*x + b*d)*a^3*c^
4*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)