3.542 \(\int x \left (a+b x^2\right )^{5/2} \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^2} \]

[Out]

((A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^2) + (B*(a + b*x^2)^(9/2))/(9*b^2)

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Rubi [A]  time = 0.100662, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (a+b x^2\right )^{7/2} (A b-a B)}{7 b^2}+\frac{B \left (a+b x^2\right )^{9/2}}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

((A*b - a*B)*(a + b*x^2)^(7/2))/(7*b^2) + (B*(a + b*x^2)^(9/2))/(9*b^2)

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Rubi in Sympy [A]  time = 13.2136, size = 37, normalized size = 0.8 \[ \frac{B \left (a + b x^{2}\right )^{\frac{9}{2}}}{9 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{7}{2}} \left (A b - B a\right )}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

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

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Mathematica [A]  time = 0.0579492, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{7/2} \left (-2 a B+9 A b+7 b B x^2\right )}{63 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^2)^(5/2)*(A + B*x^2),x]

[Out]

((a + b*x^2)^(7/2)*(9*A*b - 2*a*B + 7*b*B*x^2))/(63*b^2)

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Maple [A]  time = 0.006, size = 31, normalized size = 0.7 \[{\frac{7\,bB{x}^{2}+9\,Ab-2\,Ba}{63\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x^2+a)^(5/2)*(B*x^2+A),x)

[Out]

1/63*(b*x^2+a)^(7/2)*(7*B*b*x^2+9*A*b-2*B*a)/b^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((B*x^2 + A)*(b*x^2 + a)^(5/2)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.214506, size = 131, normalized size = 2.85 \[ \frac{{\left (7 \, B b^{4} x^{8} +{\left (19 \, B a b^{3} + 9 \, A b^{4}\right )} x^{6} - 2 \, B a^{4} + 9 \, A a^{3} b + 3 \,{\left (5 \, B a^{2} b^{2} + 9 \, A a b^{3}\right )} x^{4} +{\left (B a^{3} b + 27 \, A a^{2} b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{63 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x,x, algorithm="fricas")

[Out]

1/63*(7*B*b^4*x^8 + (19*B*a*b^3 + 9*A*b^4)*x^6 - 2*B*a^4 + 9*A*a^3*b + 3*(5*B*a^
2*b^2 + 9*A*a*b^3)*x^4 + (B*a^3*b + 27*A*a^2*b^2)*x^2)*sqrt(b*x^2 + a)/b^2

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Sympy [A]  time = 11.8294, size = 209, normalized size = 4.54 \[ \begin{cases} \frac{A a^{3} \sqrt{a + b x^{2}}}{7 b} + \frac{3 A a^{2} x^{2} \sqrt{a + b x^{2}}}{7} + \frac{3 A a b x^{4} \sqrt{a + b x^{2}}}{7} + \frac{A b^{2} x^{6} \sqrt{a + b x^{2}}}{7} - \frac{2 B a^{4} \sqrt{a + b x^{2}}}{63 b^{2}} + \frac{B a^{3} x^{2} \sqrt{a + b x^{2}}}{63 b} + \frac{5 B a^{2} x^{4} \sqrt{a + b x^{2}}}{21} + \frac{19 B a b x^{6} \sqrt{a + b x^{2}}}{63} + \frac{B b^{2} x^{8} \sqrt{a + b x^{2}}}{9} & \text{for}\: b \neq 0 \\a^{\frac{5}{2}} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x**2+a)**(5/2)*(B*x**2+A),x)

[Out]

Piecewise((A*a**3*sqrt(a + b*x**2)/(7*b) + 3*A*a**2*x**2*sqrt(a + b*x**2)/7 + 3*
A*a*b*x**4*sqrt(a + b*x**2)/7 + A*b**2*x**6*sqrt(a + b*x**2)/7 - 2*B*a**4*sqrt(a
 + b*x**2)/(63*b**2) + B*a**3*x**2*sqrt(a + b*x**2)/(63*b) + 5*B*a**2*x**4*sqrt(
a + b*x**2)/21 + 19*B*a*b*x**6*sqrt(a + b*x**2)/63 + B*b**2*x**8*sqrt(a + b*x**2
)/9, Ne(b, 0)), (a**(5/2)*(A*x**2/2 + B*x**4/4), True))

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GIAC/XCAS [A]  time = 0.236887, size = 304, normalized size = 6.61 \[ \frac{105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A a^{2} + 42 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} A a + \frac{21 \,{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B a^{2}}{b} + 3 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} A + \frac{6 \,{\left (15 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{2}\right )} B a}{b} + \frac{{\left (35 \,{\left (b x^{2} + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a^{3}\right )} B}{b}}{315 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^(5/2)*x,x, algorithm="giac")

[Out]

1/315*(105*(b*x^2 + a)^(3/2)*A*a^2 + 42*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/
2)*a)*A*a + 21*(3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*B*a^2/b + 3*(15*(b*
x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*A + 6*(15*(b
*x^2 + a)^(7/2) - 42*(b*x^2 + a)^(5/2)*a + 35*(b*x^2 + a)^(3/2)*a^2)*B*a/b + (35
*(b*x^2 + a)^(9/2) - 135*(b*x^2 + a)^(7/2)*a + 189*(b*x^2 + a)^(5/2)*a^2 - 105*(
b*x^2 + a)^(3/2)*a^3)*B/b)/b