3.1306 \(\int \frac{1}{(b d+2 c d x)^{3/2} \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=223 \[ \frac{45 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}-\frac{45 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}+\frac{90 c^2}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{9 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \sqrt{b d+2 c d x}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \sqrt{b d+2 c d x}} \]

[Out]

(90*c^2)/((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]) - 1/(2*(b^2 - 4*a*c)*d*Sqrt[b*d
+ 2*c*d*x]*(a + b*x + c*x^2)^2) + (9*c)/(2*(b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x
]*(a + b*x + c*x^2)) + (45*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*S
qrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(3/2)) - (45*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/(
(b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(3/2))

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Rubi [A]  time = 0.48868, antiderivative size = 223, 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 \[ \frac{45 c^2 \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}-\frac{45 c^2 \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{d^{3/2} \left (b^2-4 a c\right )^{13/4}}+\frac{90 c^2}{d \left (b^2-4 a c\right )^3 \sqrt{b d+2 c d x}}+\frac{9 c}{2 d \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \sqrt{b d+2 c d x}}-\frac{1}{2 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

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

[Out]

(90*c^2)/((b^2 - 4*a*c)^3*d*Sqrt[b*d + 2*c*d*x]) - 1/(2*(b^2 - 4*a*c)*d*Sqrt[b*d
+ 2*c*d*x]*(a + b*x + c*x^2)^2) + (9*c)/(2*(b^2 - 4*a*c)^2*d*Sqrt[b*d + 2*c*d*x
]*(a + b*x + c*x^2)) + (45*c^2*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 - 4*a*c)^(1/4)*S
qrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(3/2)) - (45*c^2*ArcTanh[Sqrt[d*(b + 2*c*x)]/(
(b^2 - 4*a*c)^(1/4)*Sqrt[d])])/((b^2 - 4*a*c)^(13/4)*d^(3/2))

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Rubi in Sympy [A]  time = 106.624, size = 218, normalized size = 0.98 \[ \frac{90 c^{2}}{d \left (- 4 a c + b^{2}\right )^{3} \sqrt{b d + 2 c d x}} + \frac{45 c^{2} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}}} - \frac{45 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{d^{\frac{3}{2}} \left (- 4 a c + b^{2}\right )^{\frac{13}{4}}} + \frac{9 c}{2 d \left (- 4 a c + b^{2}\right )^{2} \sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d \left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x} \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

90*c**2/(d*(-4*a*c + b**2)**3*sqrt(b*d + 2*c*d*x)) + 45*c**2*atan(sqrt(b*d + 2*c
*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(3/2)*(-4*a*c + b**2)**(13/4)) - 45*
c**2*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(d**(3/2)*(-4*a
*c + b**2)**(13/4)) + 9*c/(2*d*(-4*a*c + b**2)**2*sqrt(b*d + 2*c*d*x)*(a + b*x +
c*x**2)) - 1/(2*d*(-4*a*c + b**2)*sqrt(b*d + 2*c*d*x)*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 1.03313, size = 199, normalized size = 0.89 \[ \frac{\frac{(b+2 c x)^2 \left (-\frac{\left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac{13 c (b+2 c x)}{a+x (b+c x)}+\frac{128 c^2}{b+2 c x}\right )}{2 \left (b^2-4 a c\right )^3}+\frac{45 c^2 (b+2 c x)^{3/2} \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}-\frac{45 c^2 (b+2 c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{13/4}}}{(d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(((b + 2*c*x)^2*((128*c^2)/(b + 2*c*x) - ((b^2 - 4*a*c)*(b + 2*c*x))/(a + x*(b +
c*x))^2 + (13*c*(b + 2*c*x))/(a + x*(b + c*x))))/(2*(b^2 - 4*a*c)^3) + (45*c^2*
(b + 2*c*x)^(3/2)*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b^2 - 4*a*c)^(13
/4) - (45*c^2*(b + 2*c*x)^(3/2)*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/(b
^2 - 4*a*c)^(13/4))/(d*(b + 2*c*x))^(3/2)

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Maple [B]  time = 0.029, size = 534, normalized size = 2.4 \[ -64\,{\frac{{c}^{2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{3}\sqrt{2\,cdx+bd}}}-26\,{\frac{{c}^{2} \left ( 2\,cdx+bd \right ) ^{7/2}}{d \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-136\,{\frac{{c}^{3}d \left ( 2\,cdx+bd \right ) ^{3/2}a}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+34\,{\frac{{c}^{2}d \left ( 2\,cdx+bd \right ) ^{3/2}{b}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-{\frac{45\,{c}^{2}\sqrt{2}}{4\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{45\,{c}^{2}\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{45\,{c}^{2}\sqrt{2}}{2\,d \left ( 4\,ac-{b}^{2} \right ) ^{3}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-64*c^2/d/(4*a*c-b^2)^3/(2*c*d*x+b*d)^(1/2)-26*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)^(7/2)-136*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)^(3/2)*a+34*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)^(3/2)*b^2-45/4*c^2/d/(4*a*c-b^
2)^3/(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)))-45/2*c^2/
d/(4*a*c-b^2)^3/(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)+45/2*c^2/d/(4*a*c-b^2)^3/(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)^(3/2)*(c*x^2 + b*x + a)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.278612, size = 3374, normalized size = 15.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(180*c^4*x^4 + 360*b*c^3*x^3 - b^4 + 17*a*b^2*c + 128*a^2*c^2 + 27*(7*b^2*c^
2 + 12*a*c^3)*x^2 - 180*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*
d*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*x^3 + (b^8 -
10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d*x^2 + 2*(a*b^7 - 1
2*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*x + (a^2*b^6 - 12*a^3*b^4*c + 48*
a^4*b^2*c^2 - 64*a^5*c^3)*d)*sqrt(2*c*d*x + b*d)*(c^8/((b^26 - 52*a*b^24*c + 124
8*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 - 187432
960*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^6))^(1/4)*arctan((b^20 - 40*a*b^18*c + 720
*a^2*b^16*c^2 - 7680*a^3*b^14*c^3 + 53760*a^4*b^12*c^4 - 258048*a^5*b^10*c^5 + 8
60160*a^6*b^8*c^6 - 1966080*a^7*b^6*c^7 + 2949120*a^8*b^4*c^8 - 2621440*a^9*b^2*
c^9 + 1048576*a^10*c^10)*(c^8/((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 + 299892
736*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^6))^(3/4)*d^5/(sqrt(2*c*d*x + b*d)*c^6 + sqrt(2*c^13*d*x + b*c^12*
d + (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)*sqrt(c^8/(
(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 + 84344
832*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^6))*d^4))) - 45*(
(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d*x^4 + 2*(b^7*c - 12*a*b
^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c
^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*
c^2 - 64*a^4*b*c^3)*d*x + (a^2*b^6 - 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)
*d)*sqrt(2*c*d*x + b*d)*(c^8/((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 + 2998927
36*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^6))^(1/4)*log(91125*(b^20 - 40*a*b^18*c + 720*a^2*b^16*c^2 - 7680*a
^3*b^14*c^3 + 53760*a^4*b^12*c^4 - 258048*a^5*b^10*c^5 + 860160*a^6*b^8*c^6 - 19
66080*a^7*b^6*c^7 + 2949120*a^8*b^4*c^8 - 2621440*a^9*b^2*c^9 + 1048576*a^10*c^1
0)*(c^8/((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 - 32
7155712*a^11*b^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^6))^(3/4
)*d^5 + 91125*sqrt(2*c*d*x + b*d)*c^6) + 45*((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^
2*c^4 - 64*a^3*c^5)*d*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*
c^4)*d*x^3 + (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*
d*x^2 + 2*(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*x + (a^2*b^6
- 12*a^3*b^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*d)*sqrt(2*c*d*x + b*d)*(c^8/((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 - 327155712*a^11*b
^4*c^11 + 218103808*a^12*b^2*c^12 - 67108864*a^13*c^13)*d^6))^(1/4)*log(-91125*(
b^20 - 40*a*b^18*c + 720*a^2*b^16*c^2 - 7680*a^3*b^14*c^3 + 53760*a^4*b^12*c^4 -
258048*a^5*b^10*c^5 + 860160*a^6*b^8*c^6 - 1966080*a^7*b^6*c^7 + 2949120*a^8*b^
4*c^8 - 2621440*a^9*b^2*c^9 + 1048576*a^10*c^10)*(c^8/((b^26 - 52*a*b^24*c + 124
8*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 - 187432
960*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^6))^(3/4)*d^5 + 91125*sqrt(2*c*d*x + b*d)*
c^6) + 9*(b^3*c + 36*a*b*c^2)*x)/(((b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64
*a^3*c^5)*d*x^4 + 2*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*x^3
+ (b^8 - 10*a*b^6*c + 24*a^2*b^4*c^2 + 32*a^3*b^2*c^3 - 128*a^4*c^4)*d*x^2 + 2*
(a*b^7 - 12*a^2*b^5*c + 48*a^3*b^3*c^2 - 64*a^4*b*c^3)*d*x + (a^2*b^6 - 12*a^3*b
^4*c + 48*a^4*b^2*c^2 - 64*a^5*c^3)*d)*sqrt(2*c*d*x + b*d))

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.248678, 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)^(3/2)*(c*x^2 + b*x + a)^3),x, algorithm="giac")

[Out]

Done