3.1301 \(\int \frac{(b d+2 c d x)^{7/2}}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=170 \[ -\frac{5 c^2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c^2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.332003, 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 \[ -\frac{5 c^2 d^{7/2} \tan ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c^2 d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d (b+2 c x)}}{\sqrt{d} \sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4}}-\frac{5 c d^3 \sqrt{b d+2 c d x}}{2 \left (a+b x+c x^2\right )}-\frac{d (b d+2 c d x)^{5/2}}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 79.6786, size = 173, normalized size = 1.02 \[ - \frac{5 c^{2} d^{\frac{7}{2}} \operatorname{atan}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{4}}} - \frac{5 c^{2} d^{\frac{7}{2}} \operatorname{atanh}{\left (\frac{\sqrt{b d + 2 c d x}}{\sqrt{d} \sqrt [4]{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{4}}} - \frac{5 c d^{3} \sqrt{b d + 2 c d x}}{2 \left (a + b x + c x^{2}\right )} - \frac{d \left (b d + 2 c d x\right )^{\frac{5}{2}}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.69559, size = 160, normalized size = 0.94 \[ (d (b+2 c x))^{7/2} \left (-\frac{5 c^2 \tan ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2}}-\frac{5 c^2 \tanh ^{-1}\left (\frac{\sqrt{b+2 c x}}{\sqrt [4]{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/4} (b+2 c x)^{7/2}}-\frac{c \left (5 a+9 c x^2\right )+b^2+9 b c x}{2 (b+2 c x)^3 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(d*(b + 2*c*x))^(7/2)*(-(b^2 + 9*b*c*x + c*(5*a + 9*c*x^2))/(2*(b + 2*c*x)^3*(a
+ x*(b + c*x))^2) - (5*c^2*ArcTan[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(1/4)])/((b^2 -
4*a*c)^(3/4)*(b + 2*c*x)^(7/2)) - (5*c^2*ArcTanh[Sqrt[b + 2*c*x]/(b^2 - 4*a*c)^(
1/4)])/((b^2 - 4*a*c)^(3/4)*(b + 2*c*x)^(7/2)))

_______________________________________________________________________________________

Maple [B]  time = 0.021, size = 435, normalized size = 2.6 \[ -18\,{\frac{{c}^{2}{d}^{5} \left ( 2\,cdx+bd \right ) ^{5/2}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}-40\,{\frac{{c}^{3}{d}^{7}a\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+10\,{\frac{{c}^{2}{d}^{7}{b}^{2}\sqrt{2\,cdx+bd}}{ \left ( 4\,{c}^{2}{d}^{2}{x}^{2}+4\,bc{d}^{2}x+4\,ac{d}^{2} \right ) ^{2}}}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{4}\ln \left ({1 \left ( 2\,cdx+bd+\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) \left ( 2\,cdx+bd-\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}\sqrt{2\,cdx+bd}\sqrt{2}+\sqrt{4\,ac{d}^{2}-{b}^{2}{d}^{2}} \right ) ^{-1}} \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}+{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ({\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}}-{\frac{5\,{c}^{2}{d}^{5}\sqrt{2}}{2}\arctan \left ( -{\sqrt{2}\sqrt{2\,cdx+bd}{\frac{1}{\sqrt [4]{4\,ac{d}^{2}-{b}^{2}{d}^{2}}}}}+1 \right ) \left ( 4\,ac{d}^{2}-{b}^{2}{d}^{2} \right ) ^{-{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-18*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)^(5/2)-40*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)^(1/2)*a+10*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)^(1/2)*b^2+5/4*c^2*d^5/(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)))+5/2*c^2*d^5/(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)-5/2*c^2*d^5/(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((2*c*d*x + b*d)^(7/2)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238735, size = 832, normalized size = 4.89 \[ -\frac{20 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \arctan \left (-\frac{\left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c^{2} d^{3} + \sqrt{2 \, c^{5} d^{7} x + b c^{4} d^{7} + \sqrt{\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}}\right ) + 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c^{2} d^{3} + 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) - 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \log \left (5 \, \sqrt{2 \, c d x + b d} c^{2} d^{3} - 5 \, \left (\frac{c^{8} d^{14}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )^{\frac{1}{4}}{\left (b^{2} - 4 \, a c\right )}\right ) +{\left (9 \, c^{2} d^{3} x^{2} + 9 \, b c d^{3} x +{\left (b^{2} + 5 \, a c\right )} d^{3}\right )} \sqrt{2 \, c d x + b d}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(20*(c^8*d^14/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(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^14/(b^6 - 12
*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(b^2 - 4*a*c)/(sqrt(2*c*d*x + b*d
)*c^2*d^3 + sqrt(2*c^5*d^7*x + b*c^4*d^7 + sqrt(c^8*d^14/(b^6 - 12*a*b^4*c + 48*
a^2*b^2*c^2 - 64*a^3*c^3))*(b^4 - 8*a*b^2*c + 16*a^2*c^2)))) + 5*(c^8*d^14/(b^6
- 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(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(5*sqrt(2*c*d*x + b*d)*c^2*d^3 + 5*(c^8*d^14/(b^
6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(b^2 - 4*a*c)) - 5*(c^8*d^1
4/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(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(5*sqrt(2*c*d*x + b*d)*c^2*d^3 - 5*(c^8*d
^14/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))^(1/4)*(b^2 - 4*a*c)) + (9*
c^2*d^3*x^2 + 9*b*c*d^3*x + (b^2 + 5*a*c)*d^3)*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)

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.254419, size = 689, normalized size = 4.05 \[ -\frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} + 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} - 2 \, \sqrt{2 \, c d x + b d}\right )}}{2 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}}}\right )}{\sqrt{2} b^{2} - 4 \, \sqrt{2} a c} - \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3}{\rm ln}\left (2 \, c d x + b d + \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{2} - 4 \, \sqrt{2} a c\right )}} + \frac{5 \,{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} c^{2} d^{3}{\rm ln}\left (2 \, c d x + b d - \sqrt{2}{\left (-b^{2} d^{2} + 4 \, a c d^{2}\right )}^{\frac{1}{4}} \sqrt{2 \, c d x + b d} + \sqrt{-b^{2} d^{2} + 4 \, a c d^{2}}\right )}{2 \,{\left (\sqrt{2} b^{2} - 4 \, \sqrt{2} a c\right )}} + \frac{2 \,{\left (5 \, \sqrt{2 \, c d x + b d} b^{2} c^{2} d^{7} - 20 \, \sqrt{2 \, c d x + b d} a c^{3} d^{7} - 9 \,{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{2} d^{5}\right )}}{{\left (b^{2} d^{2} - 4 \, a c d^{2} -{\left (2 \, c d x + b d\right )}^{2}\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) - 5*(-b^2*d^2 + 4*a*c*d^2)^(1/4)*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) - 5/2*(-b^2*d^2 + 4*a*c*d^2)^(1/4)
*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) + 5/2*(-b^2*d^2
+ 4*a*c*d^2)^(1/4)*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*(5*sqrt(2*c*d*x + b*d)*b^2*c^2*d^7 - 20*sqrt(2*c*d*x + b*d)*a*c^3*d^7 -
9*(2*c*d*x + b*d)^(5/2)*c^2*d^5)/(b^2*d^2 - 4*a*c*d^2 - (2*c*d*x + b*d)^2)^2