3.1308 \(\int \frac{1}{(b d+2 c d x)^{7/2} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=256 \[ \frac{117 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac{117 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac{234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt{b d+2 c d x}}+\frac{234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac{13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}} \]

[Out]

(234*c^2)/(5*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)) + (234*c^2)/((b^2 - 4*a*c)
^4*d^3*Sqrt[b*d + 2*c*d*x]) - 1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*
x + c*x^2)^2) + (13*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x
^2)) + (117*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2
- 4*a*c)^(17/4)*d^(7/2)) - (117*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^
(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(17/4)*d^(7/2))

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Rubi [A]  time = 0.626476, antiderivative size = 256, 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 \[ \frac{117 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}-\frac{117 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{7/2} \left (b^2-4 a c\right )^{17/4}}+\frac{234 c^2}{d^3 \left (b^2-4 a c\right )^4 \sqrt{b d+2 c d x}}+\frac{234 c^2}{5 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{5/2}}+\frac{13 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{5/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

(234*c^2)/(5*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(5/2)) + (234*c^2)/((b^2 - 4*a*c)
^4*d^3*Sqrt[b*d + 2*c*d*x]) - 1/(2*(b^2 - 4*a*c)*d*(b*d + 2*c*d*x)^(5/2)*(a + b*
x + c*x^2)^2) + (13*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x
^2)) + (117*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2
- 4*a*c)^(17/4)*d^(7/2)) - (117*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^
(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(17/4)*d^(7/2))

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Rubi in Sympy [A]  time = 132.118, size = 252, normalized size = 0.98 \[ \frac{234 c^{2}}{5 d \left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} + \frac{234 c^{2}}{d^{3} \left (- 4 a c + b^{2}\right )^{4} \sqrt{b d + 2 c d x}} + \frac{117 c^{2} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{17}{4}}} - \frac{117 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{7}{2}} \left (- 4 a c + b^{2}\right )^{\frac{17}{4}}} + \frac{13 c}{2 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{5}{2}} \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

234*c**2/(5*d*(-4*a*c + b**2)**3*(b*d + 2*c*d*x)**(5/2)) + 234*c**2/(d**3*(-4*a*
c + b**2)**4*sqrt(b*d + 2*c*d*x)) + 117*c**2*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(
-4*a*c + b**2)**(1/4)))/(d**(7/2)*(-4*a*c + b**2)**(17/4)) - 117*c**2*atanh(sqrt
(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(7/2)*(-4*a*c + b**2)**(17
/4)) + 13*c/(2*d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(5/2)*(a + b*x + c*x**2)) -
1/(2*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(5/2)*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 1.52364, size = 220, normalized size = 0.86 \[ \frac{\frac{(b+2 c x)^4 \left (\frac{128 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^3}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{105 c (b+2 c x)}{a+x (b+c x)}+\frac{1920 c^2}{b+2 c x}\right )}{10 \left (b^2-4 a c\right )^4}+\frac{117 c^2 (b+2 c x)^{7/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}-\frac{117 c^2 (b+2 c x)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{17/4}}}{(d (b+2 c x))^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(7/2)*(a + b*x + c*x^2)^3),x]

[Out]

(((b + 2*c*x)^4*((128*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^3 + (1920*c^2)/(b + 2*c*x)
- (5*(b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (105*c*(b + 2*c*x))/(a + x
*(b + c*x))))/(10*(b^2 - 4*a*c)^4) + (117*c^2*(b + 2*c*x)^(7/2)*ArcTan[Sqrt[b +
2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4) - (117*c^2*(b + 2*c*x)^(7/2)*A
rcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(17/4))/(d*(b + 2*c*x
))^(7/2)

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Maple [B]  time = 0.032, size = 569, normalized size = 2.2 \[ -{\frac{64\,{c}^{2}}{5\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}}+192\,{\frac{{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}\sqrt{2\,cdx+bd}}}+42\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{7/2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+200\,{\frac{{c}^{3} \left ( 2\,cdx+bd \right ) ^{3/2}a}{d \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-50\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{3/2}{b}^{2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{117\,{c}^{2}\sqrt{2}}{4\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{117\,{c}^{2}\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{117\,{c}^{2}\sqrt{2}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(7/2)/(c*x^2+b*x+a)^3,x)

[Out]

-64/5*c^2/d/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(5/2)+192*c^2/d^3/(4*a*c-b^2)^4/(2*c*d*x
+b*d)^(1/2)+42*c^2/d^3/(4*a*c-b^2)^4/(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)+200*c^3/d/(4*a*c-b^2)^4/(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-50*c^2/d/(4*a*c-b^2)^4/(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^2+117/4*c^2/d^3/(4*a*c-b^2)^4/(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)))+117/2*c^2/d^3/(4*a*c-b^2)^4/(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)-117/2*c^2/d^3/(4*a*c-b^2)^4/(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)

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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(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.314861, size = 5211, normalized size = 20.36 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^3),x, algorithm="fricas")

[Out]

1/10*(9360*c^6*x^6 + 28080*b*c^5*x^5 - 5*b^6 + 125*a*b^4*c + 2048*a^2*b^2*c^2 -
512*a^3*c^3 + 2808*(11*b^2*c^4 + 6*a*c^5)*x^4 + 3744*(4*b^3*c^3 + 9*a*b*c^4)*x^3
+ 13*(221*b^4*c^2 + 1688*a*b^2*c^3 + 512*a^2*c^4)*x^2 + 2340*(4*(b^8*c^4 - 16*a
*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*c^3
- 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 + (1
3*b^10*c^2 - 200*a*b^8*c^3 + 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b^2*
c^6 + 2048*a^5*c^7)*d^3*x^4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 - 128
0*a^4*b^3*c^5 + 2048*a^5*b*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2 +
640*a^3*b^6*c^3 - 1920*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2 +
2*(a*b^11 - 14*a^2*b^9*c + 64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 5
12*a^6*b*c^5)*d^3*x + (a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^
3 + 256*a^6*b^2*c^4)*d^3)*sqrt(2*c*d*x + b*d)*(c^8/((b^34 - 68*a*b^32*c + 2176*a
^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 +
50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 63727
20640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 1
03817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*
c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17
*c^17)*d^14))^(1/4)*arctan(-(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*
b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 2
8114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736
*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^
13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3
+ 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 31863603
2*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048
*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159
719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^1
5 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4)/(sqrt(2*c*d*
x + b*d)*c^6 + sqrt(2*c^13*d*x + b*c^12*d + (b^18*c^8 - 36*a*b^16*c^9 + 576*a^2*
b^14*c^10 - 5376*a^3*b^12*c^11 + 32256*a^4*b^10*c^12 - 129024*a^5*b^8*c^13 + 344
064*a^6*b^6*c^14 - 589824*a^7*b^4*c^15 + 589824*a^8*b^2*c^16 - 262144*a^9*c^17)*
d^8*sqrt(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609
280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*
b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*
b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096
320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73
014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))))) - 585*(4*(b^8*c^4 - 1
6*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*
c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 +
(13*b^10*c^2 - 200*a*b^8*c^3 + 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b
^2*c^6 + 2048*a^5*c^7)*d^3*x^4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 -
1280*a^4*b^3*c^5 + 2048*a^5*b*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2
+ 640*a^3*b^6*c^3 - 1920*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2
+ 2*(a*b^11 - 14*a^2*b^9*c + 64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4
+ 512*a^6*b*c^5)*d^3*x + (a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4
*c^3 + 256*a^6*b^2*c^4)*d^3)*sqrt(2*c*d*x + b*d)*(c^8/((b^34 - 68*a*b^32*c + 217
6*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5
+ 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 63
72720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11
+ 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b
^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a
^17*c^17)*d^14))^(1/4)*log(1601613*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 183
04*a^3*b^20*c^3 + 183040*a^4*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*
c^6 - 28114944*a^7*b^12*c^7 + 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 29
9892736*a^10*b^6*c^10 - 327155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 6710
8864*a^13*c^13)*d^11*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b
^28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 3
18636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 2039
2706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^1
2 - 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*
b^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 160
1613*sqrt(2*c*d*x + b*d)*c^6) + 585*(4*(b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6
- 256*a^3*b^2*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b
^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 + (13*b^10*c^2 - 200*a*b^8*c^3
+ 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 + 2048*a^5*c^7)*d^3*x^
4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 - 1280*a^4*b^3*c^5 + 2048*a^5*b
*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2 + 640*a^3*b^6*c^3 - 1920*a^4
*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2 + 2*(a*b^11 - 14*a^2*b^9*c +
64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 512*a^6*b*c^5)*d^3*x + (a^2
*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b^2*c^4)*d^3)*
sqrt(2*c*d*x + b*d)*(c^8/((b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^
28*c^3 + 609280*a^4*b^26*c^4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 31
8636032*a^7*b^20*c^7 + 1593180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392
706048*a^10*b^14*c^10 - 51908706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12
- 159719096320*a^13*b^8*c^13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b
^4*c^15 + 73014444032*a^16*b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(1/4)*log(-1
601613*(b^26 - 52*a*b^24*c + 1248*a^2*b^22*c^2 - 18304*a^3*b^20*c^3 + 183040*a^4
*b^18*c^4 - 1317888*a^5*b^16*c^5 + 7028736*a^6*b^14*c^6 - 28114944*a^7*b^12*c^7
+ 84344832*a^8*b^10*c^8 - 187432960*a^9*b^8*c^9 + 299892736*a^10*b^6*c^10 - 3271
55712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^11*(c^8/((
b^34 - 68*a*b^32*c + 2176*a^2*b^30*c^2 - 43520*a^3*b^28*c^3 + 609280*a^4*b^26*c^
4 - 6336512*a^5*b^24*c^5 + 50692096*a^6*b^22*c^6 - 318636032*a^7*b^20*c^7 + 1593
180160*a^8*b^18*c^8 - 6372720640*a^9*b^16*c^9 + 20392706048*a^10*b^14*c^10 - 519
08706304*a^11*b^12*c^11 + 103817412608*a^12*b^10*c^12 - 159719096320*a^13*b^8*c^
13 + 182536110080*a^14*b^6*c^14 - 146028888064*a^15*b^4*c^15 + 73014444032*a^16*
b^2*c^16 - 17179869184*a^17*c^17)*d^14))^(3/4) + 1601613*sqrt(2*c*d*x + b*d)*c^6
) + 13*(5*b^5*c + 392*a*b^3*c^2 + 512*a^2*b*c^3)*x)/((4*(b^8*c^4 - 16*a*b^6*c^5
+ 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*c^3 - 16*a*b
^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 + (13*b^10*c^
2 - 200*a*b^8*c^3 + 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 + 204
8*a^5*c^7)*d^3*x^4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 - 1280*a^4*b^3
*c^5 + 2048*a^5*b*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2 + 640*a^3*b
^6*c^3 - 1920*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2 + 2*(a*b^11
- 14*a^2*b^9*c + 64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 512*a^6*b*
c^5)*d^3*x + (a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a
^6*b^2*c^4)*d^3)*sqrt(2*c*d*x + b*d))

_______________________________________________________________________________________

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(1/(2*c*d*x+b*d)**(7/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.257351, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(7/2)*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

Done