3.1303 \(\int \frac{(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=178 \[ \frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.345454, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{3 c^2 d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}+\frac{3 c^2 d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4}}-\frac{c d \sqrt{b d+2 c d x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{d \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 80.0182, size = 173, normalized size = 0.97 \[ \frac{3 c^{2} d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} + \frac{3 c^{2} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{4}}} - \frac{c d \sqrt{b d + 2 c d x}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{d \sqrt{b d + 2 c d x}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c**2*d**(3/2)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(-4*a
*c + b**2)**(7/4) + 3*c**2*d**(3/2)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c +
b**2)**(1/4)))/(-4*a*c + b**2)**(7/4) - c*d*sqrt(b*d + 2*c*d*x)/(2*(-4*a*c + b*
*2)*(a + b*x + c*x**2)) - d*sqrt(b*d + 2*c*d*x)/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.670905, size = 164, normalized size = 0.92 \[ (d (b+2 c x))^{3/2} \left (\frac{3 c^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4} (b+2 c x)^{3/2}}+\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/4} (b+2 c x)^{3/2}}+\frac{\frac{c (a+x (b+c x))}{4 a c-b^2}-1}{2 (b+2 c x) (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.021, size = 431, normalized size = 2.4 \[ 2\,{\frac{{c}^{2}{d}^{3} \left ( 2\,cdx+bd \right ) ^{5/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{c}^{2}{d}^{5}\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{16\,ac-4\,{b}^{2}}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{3\,{c}^{2}{d}^{3}\sqrt{2}}{8\,ac-2\,{b}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c^2*d^3/(4*c^2*d^2*x^2+4*b*c*d^2*x+4*a*c*d^2)^2/(4*a*c-b^2)*(2*c*d*x+b*d)^(5/2
)-6*c^2*d^5/(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)+3/4*c^2*
d^3/(4*a*c-b^2)/(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))
)+3/2*c^2*d^3/(4*a*c-b^2)/(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)-3/2*c^2*d^3/(4*a*c-b^2)/(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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243704, size = 1593, normalized size = 8.95 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(12*(c^8*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 89
60*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*(
(b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 -
2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*arctan((c^8*d^6/(b^14 - 2
8*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*
b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*(b^4 - 8*a*b^2*c + 16*a^2*c^
2)/(sqrt(2*c*d*x + b*d)*c^2*d + sqrt(2*c^5*d^3*x + b*c^4*d^3 + sqrt(c^8*d^6/(b^1
4 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504
*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))*(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)))) - 3*(c^8*d^6/(b^14 - 28*a*b^12*c + 336
*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 28672*
a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c
+ 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*
a^2*b*c)*x)*log(3*sqrt(2*c*d*x + b*d)*c^2*d + 3*(c^8*d^6/(b^14 - 28*a*b^12*c + 3
36*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4 - 21504*a^5*b^4*c^5 + 2867
2*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*(b^4 - 8*a*b^2*c + 16*a^2*c^2)) + 3*(c^8*d
^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^4
- 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*((b^2*c^2 - 4*a*
c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*
a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*log(3*sqrt(2*c*d*x + b*d)*c^2*d - 3*(c^8
*d^6/(b^14 - 28*a*b^12*c + 336*a^2*b^10*c^2 - 2240*a^3*b^8*c^3 + 8960*a^4*b^6*c^
4 - 21504*a^5*b^4*c^5 + 28672*a^6*b^2*c^6 - 16384*a^7*c^7))^(1/4)*(b^4 - 8*a*b^2
*c + 16*a^2*c^2)) + (c^2*d*x^2 + b*c*d*x + (b^2 - 3*a*c)*d)*sqrt(2*c*d*x + b*d))
/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4
- 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.24789, size = 765, normalized size = 4.3 \[ \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}} + \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{3 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{4} - 8 \, \sqrt{2} a b^{2} c + 16 \, \sqrt{2} a^{2} c^{2}\right )}} - \frac{2 \,{\left (3 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{5} - 12 \, \sqrt{2 \, c d x + b d} a c^{3} d^{5} +{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{3}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d*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^
4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) + 3*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2
*d*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^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2
)*a^2*c^2) + 3/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*c^2*d*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))/(s
qrt(2)*b^4 - 8*sqrt(2)*a*b^2*c + 16*sqrt(2)*a^2*c^2) - 3/2*(-b^2*d^2 + 4*a*c*d^2
)^(1/4)*c^2*d*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^4 - 8*sqrt(2)*a*b^2*c + 16*sq
rt(2)*a^2*c^2) - 2*(3*sqrt(2*c*d*x + b*d)*b^2*c^2*d^5 - 12*sqrt(2*c*d*x + b*d)*a
*c^3*d^5 + (2*c*d*x + b*d)^(5/2)*c^2*d^3)/((b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d
)^2)^2*(b^2 - 4*a*c))