3.4.65 \(\int \frac {(a+b x^2)^3 (A+B x^2)}{x^{5/2}} \, dx\) [365]

Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt {x}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{9} b^2 (A b+3 a B) x^{9/2}+\frac {2}{13} b^3 B x^{13/2} \]

[Out]

-2/3*a^3*A/x^(3/2)+6/5*a*b*(A*b+B*a)*x^(5/2)+2/9*b^2*(A*b+3*B*a)*x^(9/2)+2/13*b^3*B*x^(13/2)+2*a^2*(3*A*b+B*a)
*x^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {459} \begin {gather*} -\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 \sqrt {x} (a B+3 A b)+\frac {2}{9} b^2 x^{9/2} (3 a B+A b)+\frac {6}{5} a b x^{5/2} (a B+A b)+\frac {2}{13} b^3 B x^{13/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]

[Out]

(-2*a^3*A)/(3*x^(3/2)) + 2*a^2*(3*A*b + a*B)*Sqrt[x] + (6*a*b*(A*b + a*B)*x^(5/2))/5 + (2*b^2*(A*b + 3*a*B)*x^
(9/2))/9 + (2*b^3*B*x^(13/2))/13

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac {a^3 A}{x^{5/2}}+\frac {a^2 (3 A b+a B)}{\sqrt {x}}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{7/2}+b^3 B x^{11/2}\right ) \, dx\\ &=-\frac {2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt {x}+\frac {6}{5} a b (A b+a B) x^{5/2}+\frac {2}{9} b^2 (A b+3 a B) x^{9/2}+\frac {2}{13} b^3 B x^{13/2}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 78, normalized size = 0.94 \begin {gather*} \frac {-390 a^3 \left (A-3 B x^2\right )+702 a^2 b x^2 \left (5 A+B x^2\right )+78 a b^2 x^4 \left (9 A+5 B x^2\right )+10 b^3 x^6 \left (13 A+9 B x^2\right )}{585 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]

[Out]

(-390*a^3*(A - 3*B*x^2) + 702*a^2*b*x^2*(5*A + B*x^2) + 78*a*b^2*x^4*(9*A + 5*B*x^2) + 10*b^3*x^6*(13*A + 9*B*
x^2))/(585*x^(3/2))

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Maple [A]
time = 0.08, size = 78, normalized size = 0.94

method	result	size
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}+\frac {2 B a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {6 A a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {6 B \,a^{2} b \,x^{\frac {5}{2}}}{5}+6 A \,a^{2} b \sqrt {x}+2 B \,a^{3} \sqrt {x}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\)	\(78\)
default	\(\frac {2 b^{3} B \,x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}+\frac {2 B a \,b^{2} x^{\frac {9}{2}}}{3}+\frac {6 A a \,b^{2} x^{\frac {5}{2}}}{5}+\frac {6 B \,a^{2} b \,x^{\frac {5}{2}}}{5}+6 A \,a^{2} b \sqrt {x}+2 B \,a^{3} \sqrt {x}-\frac {2 a^{3} A}{3 x^{\frac {3}{2}}}\)	\(78\)
gosper	\(-\frac {2 \left (-45 B \,b^{3} x^{8}-65 x^{6} A \,b^{3}-195 x^{6} B a \,b^{2}-351 A a \,b^{2} x^{4}-351 x^{4} B \,a^{2} b -1755 x^{2} A \,a^{2} b -585 B \,a^{3} x^{2}+195 A \,a^{3}\right )}{585 x^{\frac {3}{2}}}\)	\(80\)
trager	\(-\frac {2 \left (-45 B \,b^{3} x^{8}-65 x^{6} A \,b^{3}-195 x^{6} B a \,b^{2}-351 A a \,b^{2} x^{4}-351 x^{4} B \,a^{2} b -1755 x^{2} A \,a^{2} b -585 B \,a^{3} x^{2}+195 A \,a^{3}\right )}{585 x^{\frac {3}{2}}}\)	\(80\)
risch	\(-\frac {2 \left (-45 B \,b^{3} x^{8}-65 x^{6} A \,b^{3}-195 x^{6} B a \,b^{2}-351 A a \,b^{2} x^{4}-351 x^{4} B \,a^{2} b -1755 x^{2} A \,a^{2} b -585 B \,a^{3} x^{2}+195 A \,a^{3}\right )}{585 x^{\frac {3}{2}}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/13*b^3*B*x^(13/2)+2/9*A*b^3*x^(9/2)+2/3*B*a*b^2*x^(9/2)+6/5*A*a*b^2*x^(5/2)+6/5*B*a^2*b*x^(5/2)+6*A*a^2*b*x^
(1/2)+2*B*a^3*x^(1/2)-2/3*a^3*A/x^(3/2)

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Maxima [A]
time = 0.30, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{9} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {9}{2}} + \frac {6}{5} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="maxima")

[Out]

2/13*B*b^3*x^(13/2) + 2/9*(3*B*a*b^2 + A*b^3)*x^(9/2) + 6/5*(B*a^2*b + A*a*b^2)*x^(5/2) - 2/3*A*a^3/x^(3/2) +
2*(B*a^3 + 3*A*a^2*b)*sqrt(x)

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Fricas [A]
time = 1.38, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (45 \, B b^{3} x^{8} + 65 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 351 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} - 195 \, A a^{3} + 585 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{585 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*B*b^3*x^8 + 65*(3*B*a*b^2 + A*b^3)*x^6 + 351*(B*a^2*b + A*a*b^2)*x^4 - 195*A*a^3 + 585*(B*a^3 + 3*A*
a^2*b)*x^2)/x^(3/2)

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Sympy [A]
time = 0.70, size = 110, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{3}}{3 x^{\frac {3}{2}}} + 6 A a^{2} b \sqrt {x} + \frac {6 A a b^{2} x^{\frac {5}{2}}}{5} + \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + 2 B a^{3} \sqrt {x} + \frac {6 B a^{2} b x^{\frac {5}{2}}}{5} + \frac {2 B a b^{2} x^{\frac {9}{2}}}{3} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3*(B*x**2+A)/x**(5/2),x)

[Out]

-2*A*a**3/(3*x**(3/2)) + 6*A*a**2*b*sqrt(x) + 6*A*a*b**2*x**(5/2)/5 + 2*A*b**3*x**(9/2)/9 + 2*B*a**3*sqrt(x) +
6*B*a**2*b*x**(5/2)/5 + 2*B*a*b**2*x**(9/2)/3 + 2*B*b**3*x**(13/2)/13

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Giac [A]
time = 0.64, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {6}{5} \, B a^{2} b x^{\frac {5}{2}} + \frac {6}{5} \, A a b^{2} x^{\frac {5}{2}} + 2 \, B a^{3} \sqrt {x} + 6 \, A a^{2} b \sqrt {x} - \frac {2 \, A a^{3}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="giac")

[Out]

2/13*B*b^3*x^(13/2) + 2/3*B*a*b^2*x^(9/2) + 2/9*A*b^3*x^(9/2) + 6/5*B*a^2*b*x^(5/2) + 6/5*A*a*b^2*x^(5/2) + 2*
B*a^3*sqrt(x) + 6*A*a^2*b*sqrt(x) - 2/3*A*a^3/x^(3/2)

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Mupad [B]
time = 0.02, size = 69, normalized size = 0.83 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^3+6\,A\,b\,a^2\right )+x^{9/2}\,\left (\frac {2\,A\,b^3}{9}+\frac {2\,B\,a\,b^2}{3}\right )-\frac {2\,A\,a^3}{3\,x^{3/2}}+\frac {2\,B\,b^3\,x^{13/2}}{13}+\frac {6\,a\,b\,x^{5/2}\,\left (A\,b+B\,a\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x^2)*(a + b*x^2)^3)/x^(5/2),x)

[Out]

x^(1/2)*(2*B*a^3 + 6*A*a^2*b) + x^(9/2)*((2*A*b^3)/9 + (2*B*a*b^2)/3) - (2*A*a^3)/(3*x^(3/2)) + (2*B*b^3*x^(13
/2))/13 + (6*a*b*x^(5/2)*(A*b + B*a))/5

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