3.1297 \(\int \frac{(b d+2 c d x)^{15/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=222 \[ -117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+234 c^2 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}-\frac{13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{234}{5} c^2 d^5 (b d+2 c d x)^{5/2} \]

[Out]

234*c^2*(b^2 - 4*a*c)*d^7*Sqrt[b*d + 2*c*d*x] + (234*c^2*d^5*(b*d + 2*c*d*x)^(5/
2))/5 - (d*(b*d + 2*c*d*x)^(13/2))/(2*(a + b*x + c*x^2)^2) - (13*c*d^3*(b*d + 2*
c*d*x)^(9/2))/(2*(a + b*x + c*x^2)) - 117*c^2*(b^2 - 4*a*c)^(5/4)*d^(15/2)*ArcTa
n[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 117*c^2*(b^2 - 4*a*c)^(5/
4)*d^(15/2)*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])]

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Rubi [A]  time = 0.486722, 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 \[ -117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )-117 c^2 d^{15/2} \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )+234 c^2 d^7 \left (b^2-4 a c\right ) \sqrt{b d+2 c d x}-\frac{13 c d^3 (b d+2 c d x)^{9/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{13/2}}{2 \left (a+b x+c x^2\right )^2}+\frac{234}{5} c^2 d^5 (b d+2 c d x)^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]

[Out]

234*c^2*(b^2 - 4*a*c)*d^7*Sqrt[b*d + 2*c*d*x] + (234*c^2*d^5*(b*d + 2*c*d*x)^(5/
2))/5 - (d*(b*d + 2*c*d*x)^(13/2))/(2*(a + b*x + c*x^2)^2) - (13*c*d^3*(b*d + 2*
c*d*x)^(9/2))/(2*(a + b*x + c*x^2)) - 117*c^2*(b^2 - 4*a*c)^(5/4)*d^(15/2)*ArcTa
n[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])] - 117*c^2*(b^2 - 4*a*c)^(5/
4)*d^(15/2)*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 = 114.147, size = 226, normalized size = 1.02 \[ - 117 c^{2} d^{\frac{15}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} - 117 c^{2} d^{\frac{15}{2}} \left (- 4 a c + b^{2}\right )^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )} + 234 c^{2} d^{7} \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} + \frac{234 c^{2} d^{5} \left (b d + 2 c d x\right )^{\frac{5}{2}}}{5} - \frac{13 c d^{3} \left (b d + 2 c d x\right )^{\frac{9}{2}}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{13}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(15/2)/(c*x**2+b*x+a)**3,x)

[Out]

-117*c**2*d**(15/2)*(-4*a*c + b**2)**(5/4)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4
*a*c + b**2)**(1/4))) - 117*c**2*d**(15/2)*(-4*a*c + b**2)**(5/4)*atanh(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4))) + 234*c**2*d**7*(-4*a*c + b**2)*sq
rt(b*d + 2*c*d*x) + 234*c**2*d**5*(b*d + 2*c*d*x)**(5/2)/5 - 13*c*d**3*(b*d + 2*
c*d*x)**(9/2)/(2*(a + b*x + c*x**2)) - d*(b*d + 2*c*d*x)**(13/2)/(2*(a + b*x + c
*x**2)**2)

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Mathematica [A]  time = 1.26186, size = 206, normalized size = 0.93 \[ (d (b+2 c x))^{15/2} \left (-\frac{117 c^2 \left (b^2-4 a c\right )^{5/4} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}-\frac{117 c^2 \left (b^2-4 a c\right )^{5/4} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{(b+2 c x)^{15/2}}+\frac{-512 c^2 \left (15 a c-4 b^2\right )-\frac{125 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac{5 \left (b^2-4 a c\right )^3}{(a+x (b+c x))^2}+512 b c^3 x+512 c^4 x^2}{10 (b+2 c x)^7}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(15/2)/(a + b*x + c*x^2)^3,x]

[Out]

(d*(b + 2*c*x))^(15/2)*((-512*c^2*(-4*b^2 + 15*a*c) + 512*b*c^3*x + 512*c^4*x^2
- (5*(b^2 - 4*a*c)^3)/(a + x*(b + c*x))^2 - (125*c*(b^2 - 4*a*c)^2)/(a + x*(b +
c*x)))/(10*(b + 2*c*x)^7) - (117*c^2*(b^2 - 4*a*c)^(5/4)*ArcTan[Sqrt[b + 2*c*x]/
(b^2 - 4*a*c)^(1/4)])/(b + 2*c*x)^(15/2) - (117*c^2*(b^2 - 4*a*c)^(5/4)*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 = 0.026, 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)^(15/2)/(c*x^2+b*x+a)^3,x)

[Out]

64/5*c^2*d^5*(2*c*d*x+b*d)^(5/2)-768*c^3*d^7*a*(2*c*d*x+b*d)^(1/2)+192*c^2*d^7*b
^2*(2*c*d*x+b*d)^(1/2)-800*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)^(5/2)*a^2+400*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)^(5/2)*a*b^2-50*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
)^(5/2)*b^4-2688*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)^
(1/2)*a^3+2016*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)^(1
/2)*a^2*b^2-504*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)^(
1/2)*a*b^4+42*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)^(1/
2)*b^6+936*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^2-468*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*b^2+117
/2*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)*b^4-936*c^4*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)*a^2+468*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*b^2-117/2*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)*b^4+468*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^2-234*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*b^2+117/4*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)))*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)^(15/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242878, size = 1288, normalized size = 5.8 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

-1/10*(2340*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 12
80*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(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^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 64
0*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)/((b^2*c^2 - 4*a*
c^3)*sqrt(2*c*d*x + b*d)*d^7 - sqrt(2*(b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^15*
x + (b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^15 + sqrt((b^10*c^8 - 20*a*b^8*c^9
+ 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)
))) - 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280
*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(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(-117*(b^2*c^2 - 4*a*c^3)*sqrt(2*c*d*x + b*d)*d^7 + 117*
((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c
^12 - 1024*a^5*c^13)*d^30)^(1/4)) + 585*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*
c^10 - 640*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(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(-117*(b^2*c^2 - 4*a*c^3)*
sqrt(2*c*d*x + b*d)*d^7 - 117*((b^10*c^8 - 20*a*b^8*c^9 + 160*a^2*b^6*c^10 - 640
*a^3*b^4*c^11 + 1280*a^4*b^2*c^12 - 1024*a^5*c^13)*d^30)^(1/4)) - (512*c^6*d^7*x
^6 + 1536*b*c^5*d^7*x^5 + 512*(7*b^2*c^4 - 13*a*c^5)*d^7*x^4 + 512*(9*b^3*c^3 -
26*a*b*c^4)*d^7*x^3 + 3*(641*b^4*c^2 - 520*a*b^2*c^3 - 5616*a^2*c^4)*d^7*x^2 - (
125*b^5*c - 5096*a*b^3*c^2 + 16848*a^2*b*c^3)*d^7*x - (5*b^6 + 65*a*b^4*c - 2808
*a^2*b^2*c^2 + 9360*a^3*c^3)*d^7)*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)**(15/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273444, 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)^(15/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

192*sqrt(2*c*d*x + b*d)*b^2*c^2*d^7 - 768*sqrt(2*c*d*x + b*d)*a*c^3*d^7 + 64/5*(
2*c*d*x + b*d)^(5/2)*c^2*d^5 - 117/2*sqrt(2)*(b^2*c^2*d^7 - 4*a*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)) - 117/2*sqrt(2)*(b^2*c^2*d
^7 - 4*a*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))
- 117/4*sqrt(2)*(b^2*c^2*d^7 - 4*a*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)) + 117/4*sqrt(2)*(b^2*c^2*d^7 - 4*a*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)) + 2*(21*sqrt(2*c*d*x + b*d)*b^6*c^2*d^11 -
252*sqrt(2*c*d*x + b*d)*a*b^4*c^3*d^11 + 1008*sqrt(2*c*d*x + b*d)*a^2*b^2*c^4*d
^11 - 1344*sqrt(2*c*d*x + b*d)*a^3*c^5*d^11 - 25*(2*c*d*x + b*d)^(5/2)*b^4*c^2*d
^9 + 200*(2*c*d*x + b*d)^(5/2)*a*b^2*c^3*d^9 - 400*(2*c*d*x + b*d)^(5/2)*a^2*c^4
*d^9)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2