3.160 \(\int \frac{x^6 \left (A+B x^2+C x^4+D x^6\right )}{\left (a+b x^2\right )^{9/2}} \, dx\)

Optimal. Leaf size=279 \[ \frac{x^7 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}-\frac{x \sqrt{a+b x^2} \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 a b^6}+\frac{x^3 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{12 a b^5 \sqrt{a+b x^2}}+\frac{x^5 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}} \]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^7)/(7*a*(a + b*x^2)^(7/2)) + ((b^2*B -
2*a*b*C + 3*a^2*D)*x^7)/(5*a*b^3*(a + b*x^2)^(5/2)) + ((8*b^2*B - 36*a*b*C + 99*
a^2*D)*x^5)/(60*a*b^4*(a + b*x^2)^(3/2)) + (D*x^7)/(4*b^3*(a + b*x^2)^(3/2)) + (
(8*b^2*B - 36*a*b*C + 99*a^2*D)*x^3)/(12*a*b^5*Sqrt[a + b*x^2]) - ((8*b^2*B - 36
*a*b*C + 99*a^2*D)*x*Sqrt[a + b*x^2])/(8*a*b^6) + ((8*b^2*B - 36*a*b*C + 99*a^2*
D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(13/2))

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Rubi [A]  time = 0.989022, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{x^7 \left (A-\frac{a \left (a^2 D-a b C+b^2 B\right )}{b^3}\right )}{7 a \left (a+b x^2\right )^{7/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}-\frac{x \sqrt{a+b x^2} \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 a b^6}+\frac{x^3 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{12 a b^5 \sqrt{a+b x^2}}+\frac{x^5 \left (99 a^2 D-36 a b C+8 b^2 B\right )}{60 a b^4 \left (a+b x^2\right )^{3/2}}+\frac{x^7 \left (3 a^2 D-2 a b C+b^2 B\right )}{5 a b^3 \left (a+b x^2\right )^{5/2}}+\frac{D x^7}{4 b^3 \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

((A - (a*(b^2*B - a*b*C + a^2*D))/b^3)*x^7)/(7*a*(a + b*x^2)^(7/2)) + ((b^2*B -
2*a*b*C + 3*a^2*D)*x^7)/(5*a*b^3*(a + b*x^2)^(5/2)) + ((8*b^2*B - 36*a*b*C + 99*
a^2*D)*x^5)/(60*a*b^4*(a + b*x^2)^(3/2)) + (D*x^7)/(4*b^3*(a + b*x^2)^(3/2)) + (
(8*b^2*B - 36*a*b*C + 99*a^2*D)*x^3)/(12*a*b^5*Sqrt[a + b*x^2]) - ((8*b^2*B - 36
*a*b*C + 99*a^2*D)*x*Sqrt[a + b*x^2])/(8*a*b^6) + ((8*b^2*B - 36*a*b*C + 99*a^2*
D)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(8*b^(13/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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Mathematica [A]  time = 0.384186, size = 208, normalized size = 0.75 \[ \frac{\log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right ) \left (99 a^2 D-36 a b C+8 b^2 B\right )}{8 b^{13/2}}+\frac{x \left (-10395 a^6 D+630 a^5 b \left (6 C-55 D x^2\right )-42 a^4 b^2 \left (20 B-300 C x^2+957 D x^4\right )-8 a^3 b^3 x^2 \left (350 B-1827 C x^2+2178 D x^4\right )+a^2 b^4 x^4 \left (-3248 B+6336 C x^2-1155 D x^4\right )+2 a b^5 x^6 \left (105 \left (2 C x^2+D x^4\right )-704 B\right )+120 A b^6 x^6\right )}{840 a b^6 \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(A + B*x^2 + C*x^4 + D*x^6))/(a + b*x^2)^(9/2),x]

[Out]

(x*(-10395*a^6*D + 120*A*b^6*x^6 + 630*a^5*b*(6*C - 55*D*x^2) + a^2*b^4*x^4*(-32
48*B + 6336*C*x^2 - 1155*D*x^4) - 42*a^4*b^2*(20*B - 300*C*x^2 + 957*D*x^4) - 8*
a^3*b^3*x^2*(350*B - 1827*C*x^2 + 2178*D*x^4) + 2*a*b^5*x^6*(-704*B + 105*(2*C*x
^2 + D*x^4))))/(840*a*b^6*(a + b*x^2)^(7/2)) + ((8*b^2*B - 36*a*b*C + 99*a^2*D)*
Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(8*b^(13/2))

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Maple [A]  time = 0.016, size = 460, normalized size = 1.7 \[ -{\frac{A{x}^{5}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{5\,aA{x}^{3}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{15\,{a}^{2}Ax}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{3\,aAx}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{Ax}{14\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{Ax}{7\,a{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{B{x}^{7}}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{B{x}^{5}}{5\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{B{x}^{3}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{Bx}{{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{B\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{C{x}^{9}}{2\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,aC{x}^{7}}{14\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}+{\frac{9\,aC{x}^{5}}{10\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}+{\frac{3\,aC{x}^{3}}{2\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}+{\frac{9\,Cxa}{2\,{b}^{5}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{9\,aC}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{11}{2}}}}+{\frac{D{x}^{11}}{4\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{11\,aD{x}^{9}}{8\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{99\,{a}^{2}D{x}^{7}}{56\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}}-{\frac{99\,{a}^{2}D{x}^{5}}{40\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{2}}}}-{\frac{33\,D{x}^{3}{a}^{2}}{8\,{b}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}}-{\frac{99\,Dx{a}^{2}}{8\,{b}^{6}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{99\,{a}^{2}D}{8}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(D*x^6+C*x^4+B*x^2+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)-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))+1/2*C*x^9/b/(b*x^2+a)^(7/2)+9/14*C*a/b^2*x^7/(b*x^2+a)^(7/2)+9/
10*C*a/b^3*x^5/(b*x^2+a)^(5/2)+3/2*C*a/b^4*x^3/(b*x^2+a)^(3/2)+9/2*C*a/b^5*x/(b*
x^2+a)^(1/2)-9/2*C*a/b^(11/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/4*D*x^11/b/(b*x^2+
a)^(7/2)-11/8*D*a/b^2*x^9/(b*x^2+a)^(7/2)-99/56*D*a^2/b^3*x^7/(b*x^2+a)^(7/2)-99
/40*D*a^2/b^4*x^5/(b*x^2+a)^(5/2)-33/8*D*a^2/b^5*x^3/(b*x^2+a)^(3/2)-99/8*D*a^2/
b^6*x/(b*x^2+a)^(1/2)+99/8*D*a^2/b^(13/2)*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((D*x^6 + C*x^4 + B*x^2 + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.722354, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="fricas")

[Out]

[1/1680*(2*(210*D*a*b^5*x^11 - 105*(11*D*a^2*b^4 - 4*C*a*b^5)*x^9 - 8*(2178*D*a^
3*b^3 - 792*C*a^2*b^4 + 176*B*a*b^5 - 15*A*b^6)*x^7 - 406*(99*D*a^4*b^2 - 36*C*a
^3*b^3 + 8*B*a^2*b^4)*x^5 - 350*(99*D*a^5*b - 36*C*a^4*b^2 + 8*B*a^3*b^3)*x^3 -
105*(99*D*a^6 - 36*C*a^5*b + 8*B*a^4*b^2)*x)*sqrt(b*x^2 + a)*sqrt(b) + 105*((99*
D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*b^6)*x^8 + 99*D*a^7 - 36*C*a^6*b + 8*B*a^5*b^2
+ 4*(99*D*a^4*b^3 - 36*C*a^3*b^4 + 8*B*a^2*b^5)*x^6 + 6*(99*D*a^5*b^2 - 36*C*a^4
*b^3 + 8*B*a^3*b^4)*x^4 + 4*(99*D*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*log(-
2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)))/((a*b^10*x^8 + 4*a^2*b^9*x^6 + 6
*a^3*b^8*x^4 + 4*a^4*b^7*x^2 + a^5*b^6)*sqrt(b)), 1/840*((210*D*a*b^5*x^11 - 105
*(11*D*a^2*b^4 - 4*C*a*b^5)*x^9 - 8*(2178*D*a^3*b^3 - 792*C*a^2*b^4 + 176*B*a*b^
5 - 15*A*b^6)*x^7 - 406*(99*D*a^4*b^2 - 36*C*a^3*b^3 + 8*B*a^2*b^4)*x^5 - 350*(9
9*D*a^5*b - 36*C*a^4*b^2 + 8*B*a^3*b^3)*x^3 - 105*(99*D*a^6 - 36*C*a^5*b + 8*B*a
^4*b^2)*x)*sqrt(b*x^2 + a)*sqrt(-b) + 105*((99*D*a^3*b^4 - 36*C*a^2*b^5 + 8*B*a*
b^6)*x^8 + 99*D*a^7 - 36*C*a^6*b + 8*B*a^5*b^2 + 4*(99*D*a^4*b^3 - 36*C*a^3*b^4
+ 8*B*a^2*b^5)*x^6 + 6*(99*D*a^5*b^2 - 36*C*a^4*b^3 + 8*B*a^3*b^4)*x^4 + 4*(99*D
*a^6*b - 36*C*a^5*b^2 + 8*B*a^4*b^3)*x^2)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/((
a*b^10*x^8 + 4*a^2*b^9*x^6 + 6*a^3*b^8*x^4 + 4*a^4*b^7*x^2 + a^5*b^6)*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*(D*x**6+C*x**4+B*x**2+A)/(b*x**2+a)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228934, size = 358, normalized size = 1.28 \[ \frac{{\left ({\left ({\left ({\left (105 \,{\left (\frac{2 \, D x^{2}}{b} - \frac{11 \, D a^{4} b^{9} - 4 \, C a^{3} b^{10}}{a^{3} b^{11}}\right )} x^{2} - \frac{8 \,{\left (2178 \, D a^{5} b^{8} - 792 \, C a^{4} b^{9} + 176 \, B a^{3} b^{10} - 15 \, A a^{2} b^{11}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{406 \,{\left (99 \, D a^{6} b^{7} - 36 \, C a^{5} b^{8} + 8 \, B a^{4} b^{9}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{350 \,{\left (99 \, D a^{7} b^{6} - 36 \, C a^{6} b^{7} + 8 \, B a^{5} b^{8}\right )}}{a^{3} b^{11}}\right )} x^{2} - \frac{105 \,{\left (99 \, D a^{8} b^{5} - 36 \, C a^{7} b^{6} + 8 \, B a^{6} b^{7}\right )}}{a^{3} b^{11}}\right )} x}{840 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{{\left (99 \, D a^{2} - 36 \, C a b + 8 \, B b^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{8 \, b^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^6 + C*x^4 + B*x^2 + A)*x^6/(b*x^2 + a)^(9/2),x, algorithm="giac")

[Out]

1/840*((((105*(2*D*x^2/b - (11*D*a^4*b^9 - 4*C*a^3*b^10)/(a^3*b^11))*x^2 - 8*(21
78*D*a^5*b^8 - 792*C*a^4*b^9 + 176*B*a^3*b^10 - 15*A*a^2*b^11)/(a^3*b^11))*x^2 -
406*(99*D*a^6*b^7 - 36*C*a^5*b^8 + 8*B*a^4*b^9)/(a^3*b^11))*x^2 - 350*(99*D*a^7
*b^6 - 36*C*a^6*b^7 + 8*B*a^5*b^8)/(a^3*b^11))*x^2 - 105*(99*D*a^8*b^5 - 36*C*a^
7*b^6 + 8*B*a^6*b^7)/(a^3*b^11))*x/(b*x^2 + a)^(7/2) - 1/8*(99*D*a^2 - 36*C*a*b
+ 8*B*b^2)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(13/2)