3.48 \(\int \frac{x^6 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}} \]

[Out]

-(x^6*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^4*(6*B + 7*C*x))/(35
*b^2*(a + b*x^2)^(5/2)) - (x^2*(24*B + 35*C*x))/(105*b^3*(a + b*x^2)^(3/2)) - (1
6*B + 35*C*x)/(35*b^4*Sqrt[a + b*x^2]) + (C*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]
)/b^(9/2)

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Rubi [A]  time = 0.430999, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{x^6 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac{C \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{b^{9/2}}-\frac{16 B+35 C x}{35 b^4 \sqrt{a+b x^2}}-\frac{x^2 (24 B+35 C x)}{105 b^3 \left (a+b x^2\right )^{3/2}}-\frac{x^4 (6 B+7 C x)}{35 b^2 \left (a+b x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-(x^6*(a*B - (A*b - a*C)*x))/(7*a*b*(a + b*x^2)^(7/2)) - (x^4*(6*B + 7*C*x))/(35
*b^2*(a + b*x^2)^(5/2)) - (x^2*(24*B + 35*C*x))/(105*b^3*(a + b*x^2)^(3/2)) - (1
6*B + 35*C*x)/(35*b^4*Sqrt[a + b*x^2]) + (C*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]]
)/b^(9/2)

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Rubi in Sympy [A]  time = 42.4344, size = 133, normalized size = 0.89 \[ \frac{C \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{b^{\frac{9}{2}}} - \frac{x^{4} \left (12 B + 14 C x\right )}{70 b^{2} \left (a + b x^{2}\right )^{\frac{5}{2}}} - \frac{x^{2} \left (96 B + 140 C x\right )}{420 b^{3} \left (a + b x^{2}\right )^{\frac{3}{2}}} - \frac{192 B + 420 C x}{420 b^{4} \sqrt{a + b x^{2}}} - \frac{x^{6} \left (B a - x \left (A b - C a\right )\right )}{7 a b \left (a + b x^{2}\right )^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

C*atanh(sqrt(b)*x/sqrt(a + b*x**2))/b**(9/2) - x**4*(12*B + 14*C*x)/(70*b**2*(a
+ b*x**2)**(5/2)) - x**2*(96*B + 140*C*x)/(420*b**3*(a + b*x**2)**(3/2)) - (192*
B + 420*C*x)/(420*b**4*sqrt(a + b*x**2)) - x**6*(B*a - x*(A*b - C*a))/(7*a*b*(a
+ b*x**2)**(7/2))

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Mathematica [A]  time = 0.310559, size = 127, normalized size = 0.85 \[ \frac{-3 a^4 (16 B+35 C x)-14 a^3 b x^2 (12 B+25 C x)-14 a^2 b^2 x^4 (15 B+29 C x)-a b^3 x^6 (105 B+176 C x)+15 A b^4 x^7}{105 a b^4 \left (a+b x^2\right )^{7/2}}+\frac{C \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{b^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(15*A*b^4*x^7 - 14*a^3*b*x^2*(12*B + 25*C*x) - 14*a^2*b^2*x^4*(15*B + 29*C*x) -
3*a^4*(16*B + 35*C*x) - a*b^3*x^6*(105*B + 176*C*x))/(105*a*b^4*(a + b*x^2)^(7/2
)) + (C*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/b^(9/2)

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Maple [B]  time = 0.018, size = 277, normalized size = 1.9 \[ -{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Ax}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aAx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{7\,a{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{B{x}^{6}}{b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-2\,{\frac{aB{x}^{4}}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{7/2}}}-{\frac{8\,B{x}^{2}{a}^{2}}{5\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{16\,B{a}^{3}}{35\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{C{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{C{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{C{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Cx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{C\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x)

[Out]

-1/2*A*x^5/b/(b*x^2+a)^(7/2)-5/8*A*a/b^2*x^3/(b*x^2+a)^(7/2)-15/56*A*a^2/b^3*x/(
b*x^2+a)^(7/2)+3/56*A*a/b^3*x/(b*x^2+a)^(5/2)+1/14*A/b^3*x/(b*x^2+a)^(3/2)+1/7*A
/a/b^3*x/(b*x^2+a)^(1/2)-B*x^6/b/(b*x^2+a)^(7/2)-2*B*a/b^2*x^4/(b*x^2+a)^(7/2)-8
/5*B*a^2/b^3*x^2/(b*x^2+a)^(7/2)-16/35*B*a^3/b^4/(b*x^2+a)^(7/2)-1/7*C*x^7/b/(b*
x^2+a)^(7/2)-1/5*C/b^2*x^5/(b*x^2+a)^(5/2)-1/3*C/b^3*x^3/(b*x^2+a)^(3/2)-C/b^4*x
/(b*x^2+a)^(1/2)+C/b^(9/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))

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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((C*x^2 + B*x + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28308, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (105 \, B a b^{3} x^{6} + 406 \, C a^{2} b^{2} x^{5} + 210 \, B a^{2} b^{2} x^{4} + 350 \, C a^{3} b x^{3} +{\left (176 \, C a b^{3} - 15 \, A b^{4}\right )} x^{7} + 168 \, B a^{3} b x^{2} + 105 \, C a^{4} x + 48 \, B a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{b} - 105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{210 \,{\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} \sqrt{b}}, -\frac{{\left (105 \, B a b^{3} x^{6} + 406 \, C a^{2} b^{2} x^{5} + 210 \, B a^{2} b^{2} x^{4} + 350 \, C a^{3} b x^{3} +{\left (176 \, C a b^{3} - 15 \, A b^{4}\right )} x^{7} + 168 \, B a^{3} b x^{2} + 105 \, C a^{4} x + 48 \, B a^{4}\right )} \sqrt{b x^{2} + a} \sqrt{-b} - 105 \,{\left (C a b^{4} x^{8} + 4 \, C a^{2} b^{3} x^{6} + 6 \, C a^{3} b^{2} x^{4} + 4 \, C a^{4} b x^{2} + C a^{5}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{105 \,{\left (a b^{8} x^{8} + 4 \, a^{2} b^{7} x^{6} + 6 \, a^{3} b^{6} x^{4} + 4 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )} \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[-1/210*(2*(105*B*a*b^3*x^6 + 406*C*a^2*b^2*x^5 + 210*B*a^2*b^2*x^4 + 350*C*a^3*
b*x^3 + (176*C*a*b^3 - 15*A*b^4)*x^7 + 168*B*a^3*b*x^2 + 105*C*a^4*x + 48*B*a^4)
*sqrt(b*x^2 + a)*sqrt(b) - 105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C*a^3*b^2*x^4
+ 4*C*a^4*b*x^2 + C*a^5)*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((
a*b^8*x^8 + 4*a^2*b^7*x^6 + 6*a^3*b^6*x^4 + 4*a^4*b^5*x^2 + a^5*b^4)*sqrt(b)), -
1/105*((105*B*a*b^3*x^6 + 406*C*a^2*b^2*x^5 + 210*B*a^2*b^2*x^4 + 350*C*a^3*b*x^
3 + (176*C*a*b^3 - 15*A*b^4)*x^7 + 168*B*a^3*b*x^2 + 105*C*a^4*x + 48*B*a^4)*sqr
t(b*x^2 + a)*sqrt(-b) - 105*(C*a*b^4*x^8 + 4*C*a^2*b^3*x^6 + 6*C*a^3*b^2*x^4 + 4
*C*a^4*b*x^2 + C*a^5)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((a*b^8*x^8 + 4*a^2*b^
7*x^6 + 6*a^3*b^6*x^4 + 4*a^4*b^5*x^2 + a^5*b^4)*sqrt(-b))]

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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(x**6*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221754, size = 186, normalized size = 1.24 \[ -\frac{{\left ({\left ({\left ({\left ({\left (x{\left (\frac{105 \, B}{b} + \frac{{\left (176 \, C a^{3} b^{7} - 15 \, A a^{2} b^{8}\right )} x}{a^{3} b^{8}}\right )} + \frac{406 \, C a}{b^{2}}\right )} x + \frac{210 \, B a}{b^{2}}\right )} x + \frac{350 \, C a^{2}}{b^{3}}\right )} x + \frac{168 \, B a^{2}}{b^{3}}\right )} x + \frac{105 \, C a^{3}}{b^{4}}\right )} x + \frac{48 \, B a^{3}}{b^{4}}}{105 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{C{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((C*x^2 + B*x + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

-1/105*((((((x*(105*B/b + (176*C*a^3*b^7 - 15*A*a^2*b^8)*x/(a^3*b^8)) + 406*C*a/
b^2)*x + 210*B*a/b^2)*x + 350*C*a^2/b^3)*x + 168*B*a^2/b^3)*x + 105*C*a^3/b^4)*x
+ 48*B*a^3/b^4)/(b*x^2 + a)^(7/2) - C*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(
9/2)