3.1307 \(\int \frac{1}{(b d+2 c d x)^{5/2} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=225 \[ -\frac{77 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}-\frac{77 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}+\frac{154 c^2}{3 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{11 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{3/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{3/2}} \]

[Out]

(154*c^2)/(3*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) - 1/(2*(b^2 - 4*a*c)*d*(b*
d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2) + (11*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c
*d*x)^(3/2)*(a + b*x + c*x^2)) - (77*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2)) - (77*c^2*ArcTanh[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.502267, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{77 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}-\frac{77 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{5/2} \left (b^2-4 a c\right )^{15/4}}+\frac{154 c^2}{3 d \left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}+\frac{11 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) (b d+2 c d x)^{3/2}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 (b d+2 c d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x]

[Out]

(154*c^2)/(3*(b^2 - 4*a*c)^3*d*(b*d + 2*c*d*x)^(3/2)) - 1/(2*(b^2 - 4*a*c)*d*(b*
d + 2*c*d*x)^(3/2)*(a + b*x + c*x^2)^2) + (11*c)/(2*(b^2 - 4*a*c)^2*d*(b*d + 2*c
*d*x)^(3/2)*(a + b*x + c*x^2)) - (77*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*
c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2)) - (77*c^2*ArcTanh[Sqrt[d*(b +
2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(15/4)*d^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 106.79, size = 219, normalized size = 0.97 \[ \frac{154 c^{2}}{3 d \left (- 4 a c + b^{2}\right )^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{77 c^{2} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}}} - \frac{77 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{5}{2}} \left (- 4 a c + b^{2}\right )^{\frac{15}{4}}} + \frac{11 c}{2 d \left (- 4 a c + b^{2}\right )^{2} \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d \left (- 4 a c + b^{2}\right ) \left (b d + 2 c d x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**(5/2)/(c*x**2+b*x+a)**3,x)

[Out]

154*c**2/(3*d*(-4*a*c + b**2)**3*(b*d + 2*c*d*x)**(3/2)) - 77*c**2*atan(sqrt(b*d
+ 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(5/2)*(-4*a*c + b**2)**(15/4))
- 77*c**2*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(5/2)
*(-4*a*c + b**2)**(15/4)) + 11*c/(2*d*(-4*a*c + b**2)**2*(b*d + 2*c*d*x)**(3/2)*
(a + b*x + c*x**2)) - 1/(2*d*(-4*a*c + b**2)*(b*d + 2*c*d*x)**(3/2)*(a + b*x + c
*x**2)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 1.111, size = 187, normalized size = 0.83 \[ \frac{\frac{(b+2 c x)^3 \left (-\frac{3 \left (b^2-4 a c\right )}{(a+x (b+c x))^2}+\frac{45 c}{a+x (b+c x)}+\frac{128 c^2}{(b+2 c x)^2}\right )}{6 \left (b^2-4 a c\right )^3}-\frac{77 c^2 (b+2 c x)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{15/4}}-\frac{77 c^2 (b+2 c x)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{15/4}}}{(d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^(5/2)*(a + b*x + c*x^2)^3),x]

[Out]

(((b + 2*c*x)^3*((128*c^2)/(b + 2*c*x)^2 - (3*(b^2 - 4*a*c))/(a + x*(b + c*x))^2
+ (45*c)/(a + x*(b + c*x))))/(6*(b^2 - 4*a*c)^3) - (77*c^2*(b + 2*c*x)^(5/2)*Ar
cTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(15/4) - (77*c^2*(b + 2
*c*x)^(5/2)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(15/4))/
(d*(b + 2*c*x))^(5/2)

_______________________________________________________________________________________

Maple [B]  time = 0.026, size = 534, normalized size = 2.4 \[ -{\frac{64\,{c}^{2}}{3\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}}-30\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{5/2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-152\,{\frac{{c}^{3}d\sqrt{2\,cdx+bd}a}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+38\,{\frac{{c}^{2}d\sqrt{2\,cdx+bd}{b}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-{\frac{77\,{c}^{2}\sqrt{2}}{4\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{77\,{c}^{2}\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{77\,{c}^{2}\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^(5/2)/(c*x^2+b*x+a)^3,x)

[Out]

-64/3*c^2/d/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(3/2)-30*c^2/d/(4*a*c-b^2)^3/(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)-152*c^3*d/(4*a*c-b^2)^3/(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+38*c^2*d/(4*a*c-b^2)^3/(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^2-77/4*c^2/d/(4*a*c-
b^2)^3/(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)))-77/2*c^
2/d/(4*a*c-b^2)^3/(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)+77/2*c^2/d/(4*a*c-b^2)^3/(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)

_______________________________________________________________________________________

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)^(5/2)*(c*x^2 + b*x + a)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.278712, size = 3744, normalized size = 16.64 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^3),x, algorithm="fricas")

[Out]

1/6*(308*c^4*x^4 + 616*b*c^3*x^3 - 3*b^4 + 57*a*b^2*c + 128*a^2*c^2 + 11*(31*b^2
*c^2 + 44*a*c^3)*x^2 + 924*(2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*
c^6)*d^2*x^5 + 5*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^
4 + 4*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*
x^3 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x
^2 + 2*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2
*x + (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*d^2)*sqrt(2*c*d*x
+ b*d)*(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 3494
40*a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b
^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^1
0*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b
^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)*arctan((
b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3*(c^8/((b^
30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4
- 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724
160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224
960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 402653
1840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)/(sqrt(2*c*d*x + b*d)*c^2
+ sqrt((b^16 - 32*a*b^14*c + 448*a^2*b^12*c^2 - 3584*a^3*b^10*c^3 + 17920*a^4*b
^8*c^4 - 57344*a^5*b^6*c^5 + 114688*a^6*b^4*c^6 - 131072*a^7*b^2*c^7 + 65536*a^8
*c^8)*d^6*sqrt(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3
+ 349440*a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 10543104
0*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a
^10*b^10*c^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720
*a^13*b^4*c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10)) + 2*c^5
*d*x + b*c^4*d))) - 231*(2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6
)*d^2*x^5 + 5*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^4 +
4*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*x^3
+ (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x^2
+ 2*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*x
+ (a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*d^2)*sqrt(2*c*d*x + b
*d)*(c^8/((b^30 - 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*
a^4*b^22*c^4 - 3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16
*c^7 + 421724160*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c
^10 - 5725224960*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*
c^13 + 4026531840*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)*log(77*(b^8
- 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3*(c^8/((b^30
- 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3
075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160
*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960
*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 402653184
0*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4) + 77*sqrt(2*c*d*x + b*d)*c^
2) + 231*(2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d^2*x^5 + 5*(
b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^4 + 4*(b^8*c - 11*
a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*x^3 + (b^9 - 6*a*b
^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x^2 + 2*(a*b^8 - 11
*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*x + (a^2*b^7 - 12
*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*d^2)*sqrt(2*c*d*x + b*d)*(c^8/((b^30
- 60*a*b^28*c + 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 -
3075072*a^5*b^20*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 42172416
0*a^8*b^14*c^8 - 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 572522496
0*a^11*b^8*c^11 + 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 40265318
40*a^14*b^2*c^14 - 1073741824*a^15*c^15)*d^10))^(1/4)*log(-77*(b^8 - 16*a*b^6*c
+ 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3*(c^8/((b^30 - 60*a*b^28*c
+ 1680*a^2*b^26*c^2 - 29120*a^3*b^24*c^3 + 349440*a^4*b^22*c^4 - 3075072*a^5*b^2
0*c^5 + 20500480*a^6*b^18*c^6 - 105431040*a^7*b^16*c^7 + 421724160*a^8*b^14*c^8
- 1312030720*a^9*b^12*c^9 + 3148873728*a^10*b^10*c^10 - 5725224960*a^11*b^8*c^11
+ 7633633280*a^12*b^6*c^12 - 7046430720*a^13*b^4*c^13 + 4026531840*a^14*b^2*c^1
4 - 1073741824*a^15*c^15)*d^10))^(1/4) + 77*sqrt(2*c*d*x + b*d)*c^2) + 11*(3*b^3
*c + 44*a*b*c^2)*x)/((2*(b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*d
^2*x^5 + 5*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*d^2*x^4 + 4*
(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*d^2*x^3 +
(b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*d^2*x^2 + 2
*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*d^2*x + (
a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*d^2)*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)**(5/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.246353, size = 1098, normalized size = 4.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^(5/2)*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

-77*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*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^
8*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*d^3 - 256*sqrt(2)*a^3*b^
2*c^3*d^3 + 256*sqrt(2)*a^4*c^4*d^3) - 77*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*arcta
n(-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^8*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(
2)*a^2*b^4*c^2*d^3 - 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4*d^3) - 77
/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*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^8*d^3 -
16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c^2*d^3 - 256*sqrt(2)*a^3*b^2*c^3*d
^3 + 256*sqrt(2)*a^4*c^4*d^3) + 77/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*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^8*d^3 - 16*sqrt(2)*a*b^6*c*d^3 + 96*sqrt(2)*a^2*b^4*c
^2*d^3 - 256*sqrt(2)*a^3*b^2*c^3*d^3 + 256*sqrt(2)*a^4*c^4*d^3) + 64/3*c^2/((b^6
*d - 12*a*b^4*c*d + 48*a^2*b^2*c^2*d - 64*a^3*c^3*d)*(2*c*d*x + b*d)^(3/2)) - 2*
(19*sqrt(2*c*d*x + b*d)*b^2*c^2*d^2 - 76*sqrt(2*c*d*x + b*d)*a*c^3*d^2 - 15*(2*c
*d*x + b*d)^(5/2)*c^2)/((b^6*d - 12*a*b^4*c*d + 48*a^2*b^2*c^2*d - 64*a^3*c^3*d)
*(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2)