3.1300 \(\int \frac{(b d+2 c d x)^{9/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=170 \[ \frac{21 c^2 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{21 c^2 d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(d*(b*d + 2*c*d*x)^(7/2))/(2*(a + b*x + c*x^2)^2) - (7*c*d^3*(b*d + 2*c*d*x)^(3
/2))/(2*(a + b*x + c*x^2)) + (21*c^2*d^(9/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 -
4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4) - (21*c^2*d^(9/2)*ArcTanh[Sqrt[d*(b
+ 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4)

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Rubi [A]  time = 0.342719, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{21 c^2 d^{9/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{21 c^2 d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c}}-\frac{7 c d^3 (b d+2 c d x)^{3/2}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{7/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x]

[Out]

-(d*(b*d + 2*c*d*x)^(7/2))/(2*(a + b*x + c*x^2)^2) - (7*c*d^3*(b*d + 2*c*d*x)^(3
/2))/(2*(a + b*x + c*x^2)) + (21*c^2*d^(9/2)*ArcTan[Sqrt[d*(b + 2*c*x)]/((b^2 -
4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4) - (21*c^2*d^(9/2)*ArcTanh[Sqrt[d*(b
+ 2*c*x)]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])])/(b^2 - 4*a*c)^(1/4)

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Rubi in Sympy [A]  time = 79.5113, size = 172, normalized size = 1.01 \[ \frac{21 c^{2} d^{\frac{9}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} - \frac{21 c^{2} d^{\frac{9}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\sqrt [4]{- 4 a c + b^{2}}} - \frac{7 c d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{7}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**(9/2)/(c*x**2+b*x+a)**3,x)

[Out]

21*c**2*d**(9/2)*atan(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c + b**2)**(1/4)))/(-4*
a*c + b**2)**(1/4) - 21*c**2*d**(9/2)*atanh(sqrt(b*d + 2*c*d*x)/(sqrt(d)*(-4*a*c
+ b**2)**(1/4)))/(-4*a*c + b**2)**(1/4) - 7*c*d**3*(b*d + 2*c*d*x)**(3/2)/(2*(a
+ b*x + c*x**2)) - d*(b*d + 2*c*d*x)**(7/2)/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.71129, size = 160, normalized size = 0.94 \[ (d (b+2 c x))^{9/2} \left (\frac{21 c^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{9/2}}-\frac{21 c^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\sqrt [4]{b^2-4 a c} (b+2 c x)^{9/2}}-\frac{c \left (7 a+11 c x^2\right )+b^2+11 b c x}{2 (b+2 c x)^3 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^(9/2)/(a + b*x + c*x^2)^3,x]

[Out]

(d*(b + 2*c*x))^(9/2)*(-(b^2 + 11*b*c*x + c*(7*a + 11*c*x^2))/(2*(b + 2*c*x)^3*(
a + x*(b + c*x))^2) + (21*c^2*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/((b^2
- 4*a*c)^(1/4)*(b + 2*c*x)^(9/2)) - (21*c^2*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*
c)^(1/4)])/((b^2 - 4*a*c)^(1/4)*(b + 2*c*x)^(9/2)))

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Maple [B]  time = 0.02, size = 435, normalized size = 2.6 \[ -22\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{7/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-56\,{\frac{{c}^{3}{d}^{7} \left ( 2\,cdx+bd \right ) ^{3/2}a}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+14\,{\frac{{c}^{2}{d}^{7}{b}^{2} \left ( 2\,cdx+bd \right ) ^{3/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{4}\ln \left ({1 \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}-{\frac{21\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^(9/2)/(c*x^2+b*x+a)^3,x)

[Out]

-22*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)^(7/2)-56*c^3*d
^7/(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+14*c^2*d^7/(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+21/4*c^2*d^5/(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)))+21/2*c^2*d^5/(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)-21/2*c^2*d^5/(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((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237182, size = 653, normalized size = 3.84 \[ \frac{84 \, \left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \arctan \left (-\frac{\left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c^{6} d^{13} + \sqrt{2 \, c^{13} d^{27} x + b c^{12} d^{27} + \sqrt{\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}}{\left (b^{2} c^{8} - 4 \, a c^{9}\right )} d^{18}}}\right ) - 21 \, \left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (9261 \, \sqrt{2 \, c d x + b d} c^{6} d^{13} + 9261 \, \left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) + 21 \, \left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (9261 \, \sqrt{2 \, c d x + b d} c^{6} d^{13} - 9261 \, \left (\frac{c^{8} d^{18}}{b^{2} - 4 \, a c}\right )^{\frac{3}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) -{\left (22 \, c^{3} d^{4} x^{3} + 33 \, b c^{2} d^{4} x^{2} +{\left (13 \, b^{2} c + 14 \, a c^{2}\right )} d^{4} x +{\left (b^{3} + 7 \, a b c\right )} d^{4}\right )} \sqrt{2 \, c d x + b d}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

1/2*(84*(c^8*d^18/(b^2 - 4*a*c))^(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(-(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a*c)/(sqrt(2*c*
d*x + b*d)*c^6*d^13 + sqrt(2*c^13*d^27*x + b*c^12*d^27 + sqrt(c^8*d^18/(b^2 - 4*
a*c))*(b^2*c^8 - 4*a*c^9)*d^18))) - 21*(c^8*d^18/(b^2 - 4*a*c))^(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(9261*sqrt(2*c*d*x + b*d)*c^6
*d^13 + 9261*(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a*c)) + 21*(c^8*d^18/(b^2 -
4*a*c))^(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(926
1*sqrt(2*c*d*x + b*d)*c^6*d^13 - 9261*(c^8*d^18/(b^2 - 4*a*c))^(3/4)*(b^2 - 4*a*
c)) - (22*c^3*d^4*x^3 + 33*b*c^2*d^4*x^2 + (13*b^2*c + 14*a*c^2)*d^4*x + (b^3 +
7*a*b*c)*d^4)*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)**(9/2)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.262272, size = 689, normalized size = 4.05 \[ -\frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{3} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{3} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} + \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{3}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{2} - 4 \, \sqrt{2} a c\right )}} - \frac{21 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{3}{4}} c^{2} d^{3}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{2} - 4 \, \sqrt{2} a c\right )}} + \frac{2 \,{\left (7 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} b^{2} c^{2} d^{7} - 28 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} a c^{3} d^{7} - 11 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{5}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^(9/2)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-21*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3*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^2 - 4*sqrt(2)*a*c) - 21*(-b^2*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3*arctan(-1/2*sqr
t(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^2 - 4*sqrt(2)*a*c) + 21/2*(-b^2*d^2 + 4*a*c*d^2)^(3
/4)*c^2*d^3*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^2 - 4*sqrt(2)*a*c) - 21/2*(-b^2
*d^2 + 4*a*c*d^2)^(3/4)*c^2*d^3*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^2 - 4*sqrt(
2)*a*c) + 2*(7*(2*c*d*x + b*d)^(3/2)*b^2*c^2*d^7 - 28*(2*c*d*x + b*d)^(3/2)*a*c^
3*d^7 - 11*(2*c*d*x + b*d)^(7/2)*c^2*d^5)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)
^2)^2