3.47 \(\int \frac{x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=213 \[ -\frac{16 \sqrt{a+b x^2} (A b-8 a C)}{35 a b^5}-\frac{x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt{a+b x^2}}-\frac{x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

[Out]

-(x^7*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^5*(7*a*B - (A*b - 8*
a*C)*x))/(35*a*b^2*(a + b*x^2)^(5/2)) - (x^3*(35*a*B - 6*(A*b - 8*a*C)*x))/(105*
a*b^3*(a + b*x^2)^(3/2)) - (x*(35*a*B - 8*(A*b - 8*a*C)*x))/(35*a*b^4*Sqrt[a + b
*x^2]) - (16*(A*b - 8*a*C)*Sqrt[a + b*x^2])/(35*a*b^5) + (B*ArcTanh[(Sqrt[b]*x)/
Sqrt[a + b*x^2]])/b^(9/2)

_______________________________________________________________________________________

Rubi [A]  time = 0.818364, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{16 \sqrt{a+b x^2} (A b-8 a C)}{35 a b^5}-\frac{x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt{a+b x^2}}-\frac{x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac{x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{B \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(x^7*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^5*(7*a*B - (A*b - 8*
a*C)*x))/(35*a*b^2*(a + b*x^2)^(5/2)) - (x^3*(35*a*B - 6*(A*b - 8*a*C)*x))/(105*
a*b^3*(a + b*x^2)^(3/2)) - (x*(35*a*B - 8*(A*b - 8*a*C)*x))/(35*a*b^4*Sqrt[a + b
*x^2]) - (16*(A*b - 8*a*C)*Sqrt[a + b*x^2])/(35*a*b^5) + (B*ArcTanh[(Sqrt[b]*x)/
Sqrt[a + b*x^2]])/b^(9/2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 59.1647, size = 192, normalized size = 0.9 \[ \frac{B \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{9}{2}}} - \frac{x^{7} \left (B a - x \left (A b - C a\right )\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} - \frac{x^{5} \left (14 B a - x \left (2 A b - 16 C a\right )\right )}{70 a b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{x^{3} \left (140 B a - x \left (24 A b - 192 C a\right )\right )}{420 a b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{x \left (840 B a - x \left (192 A b - 1536 C a\right )\right )}{840 a b^{4} \sqrt{a + b x^{2}}} - \frac{16 \sqrt{a + b x^{2}} \left (A b - 8 C a\right )}{35 a b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

B*atanh(sqrt(b)*x/sqrt(a + b*x**2))/b**(9/2) - x**7*(B*a - x*(A*b - C*a))/(7*a*b
*(a + b*x**2)**(7/2)) - x**5*(14*B*a - x*(2*A*b - 16*C*a))/(70*a*b**2*(a + b*x**
2)**(5/2)) - x**3*(140*B*a - x*(24*A*b - 192*C*a))/(420*a*b**3*(a + b*x**2)**(3/
2)) - x*(840*B*a - x*(192*A*b - 1536*C*a))/(840*a*b**4*sqrt(a + b*x**2)) - 16*sq
rt(a + b*x**2)*(A*b - 8*C*a)/(35*a*b**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.299105, size = 157, normalized size = 0.74 \[ \frac{384 a^4 C-3 a^3 b (16 A+7 x (5 B-64 C x))+14 a^2 b^2 x^2 (5 x (24 C x-5 B)-12 A)+14 a b^3 x^4 (x (60 C x-29 B)-15 A)+105 \sqrt{b} B \left (a+b x^2\right )^{7/2} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+b^4 x^6 (x (105 C x-176 B)-105 A)}{105 b^5 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(384*a^4*C - 3*a^3*b*(16*A + 7*x*(5*B - 64*C*x)) + 14*a^2*b^2*x^2*(-12*A + 5*x*(
-5*B + 24*C*x)) + 14*a*b^3*x^4*(-15*A + x*(-29*B + 60*C*x)) + b^4*x^6*(-105*A +
x*(-176*B + 105*C*x)) + 105*Sqrt[b]*B*(a + b*x^2)^(7/2)*Log[b*x + Sqrt[b]*Sqrt[a
+ b*x^2]])/(105*b^5*(a + b*x^2)^(7/2))

_______________________________________________________________________________________

Maple [A]  time = 0.049, size = 265, normalized size = 1.2 \[ -{\frac{A{x}^{6}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-2\,{\frac{aA{x}^{4}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}-{\frac{8\,A{a}^{2}{x}^{2}}{5\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{16\,A{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{B{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{C{x}^{8}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+8\,{\frac{aC{x}^{6}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}+16\,{\frac{{a}^{2}C{x}^{4}}{{b}^{3} \left ( b{x}^{2}+a \right ) ^{7/2}}}+{\frac{64\,C{a}^{3}{x}^{2}}{5\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{128\,C{a}^{4}}{35\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

-A*x^6/b/(b*x^2+a)^(7/2)-2*A*a/b^2*x^4/(b*x^2+a)^(7/2)-8/5*A*a^2/b^3*x^2/(b*x^2+
a)^(7/2)-16/35*A*a^3/b^4/(b*x^2+a)^(7/2)-1/7*B*x^7/b/(b*x^2+a)^(7/2)-1/5*B/b^2*x
^5/(b*x^2+a)^(5/2)-1/3*B/b^3*x^3/(b*x^2+a)^(3/2)-B/b^4*x/(b*x^2+a)^(1/2)+B/b^(9/
2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+C*x^8/b/(b*x^2+a)^(7/2)+8*C*a/b^2*x^6/(b*x^2+a)
^(7/2)+16*C*a^2/b^3*x^4/(b*x^2+a)^(7/2)+64/5*C*a^3/b^4*x^2/(b*x^2+a)^(7/2)+128/3
5*C*a^4/b^5/(b*x^2+a)^(7/2)

_______________________________________________________________________________________

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((C*x^2 + B*x + A)*x^7/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.292743, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \,{\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \,{\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \,{\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{b} + 105 \,{\left (B b^{5} x^{8} + 4 \, B a b^{4} x^{6} + 6 \, B a^{2} b^{3} x^{4} + 4 \, B a^{3} b^{2} x^{2} + B a^{4} b\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{210 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \sqrt{b}}, \frac{{\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \,{\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \,{\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \,{\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 105 \,{\left (B b^{5} x^{8} + 4 \, B a b^{4} x^{6} + 6 \, B a^{2} b^{3} x^{4} + 4 \, B a^{3} b^{2} x^{2} + B a^{4} b\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{105 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^7/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(2*(105*C*b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 +
105*(8*C*a*b^3 - A*b^4)*x^6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C
*a^2*b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)*sqrt(
b) + 105*(B*b^5*x^8 + 4*B*a*b^4*x^6 + 6*B*a^2*b^3*x^4 + 4*B*a^3*b^2*x^2 + B*a^4*
b)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((b^9*x^8 + 4*a*b^8*x^6
+ 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*sqrt(b)), 1/105*((105*C*b^4*x^8 - 176
*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*C*a*b^3 - A*b^4)*x^6 -
105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 168*
(8*C*a^3*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)*sqrt(-b) + 105*(B*b^5*x^8 + 4*B*a*b
^4*x^6 + 6*B*a^2*b^3*x^4 + 4*B*a^3*b^2*x^2 + B*a^4*b)*arctan(sqrt(-b)*x/sqrt(b*x
^2 + a)))/((b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)*sqr
t(-b))]

_______________________________________________________________________________________

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(x**7*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227557, size = 275, normalized size = 1.29 \[ \frac{{\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac{105 \, C x}{b} - \frac{176 \, B}{b}\right )} x + \frac{105 \,{\left (8 \, C a^{4} b^{7} - A a^{3} b^{8}\right )}}{a^{3} b^{9}}\right )} x - \frac{406 \, B a}{b^{2}}\right )} x + \frac{210 \,{\left (8 \, C a^{5} b^{6} - A a^{4} b^{7}\right )}}{a^{3} b^{9}}\right )} x - \frac{350 \, B a^{2}}{b^{3}}\right )} x + \frac{168 \,{\left (8 \, C a^{6} b^{5} - A a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x - \frac{105 \, B a^{3}}{b^{4}}\right )} x + \frac{48 \,{\left (8 \, C a^{7} b^{4} - A a^{6} b^{5}\right )}}{a^{3} b^{9}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{B{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^7/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

1/105*((((((((105*C*x/b - 176*B/b)*x + 105*(8*C*a^4*b^7 - A*a^3*b^8)/(a^3*b^9))*
x - 406*B*a/b^2)*x + 210*(8*C*a^5*b^6 - A*a^4*b^7)/(a^3*b^9))*x - 350*B*a^2/b^3)
*x + 168*(8*C*a^6*b^5 - A*a^5*b^6)/(a^3*b^9))*x - 105*B*a^3/b^4)*x + 48*(8*C*a^7
*b^4 - A*a^6*b^5)/(a^3*b^9))/(b*x^2 + a)^(7/2) - B*ln(abs(-sqrt(b)*x + sqrt(b*x^
2 + a)))/b^(9/2)