3.4.61 \(\int x^{3/2} (a+b x^2)^3 (A+B x^2) \, dx\) [361]

Optimal. Leaf size=85 \[ \frac {2}{5} a^3 A x^{5/2}+\frac {2}{9} a^2 (3 A b+a B) x^{9/2}+\frac {6}{13} a b (A b+a B) x^{13/2}+\frac {2}{17} b^2 (A b+3 a B) x^{17/2}+\frac {2}{21} b^3 B x^{21/2} \]

[Out]

2/5*a^3*A*x^(5/2)+2/9*a^2*(3*A*b+B*a)*x^(9/2)+6/13*a*b*(A*b+B*a)*x^(13/2)+2/17*b^2*(A*b+3*B*a)*x^(17/2)+2/21*b
^3*B*x^(21/2)

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Rubi [A]
time = 0.03, antiderivative size = 85, 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}{5} a^3 A x^{5/2}+\frac {2}{9} a^2 x^{9/2} (a B+3 A b)+\frac {2}{17} b^2 x^{17/2} (3 a B+A b)+\frac {6}{13} a b x^{13/2} (a B+A b)+\frac {2}{21} b^3 B x^{21/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^3*A*x^(5/2))/5 + (2*a^2*(3*A*b + a*B)*x^(9/2))/9 + (6*a*b*(A*b + a*B)*x^(13/2))/13 + (2*b^2*(A*b + 3*a*B)
*x^(17/2))/17 + (2*b^3*B*x^(21/2))/21

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 x^{3/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^{3/2}+a^2 (3 A b+a B) x^{7/2}+3 a b (A b+a B) x^{11/2}+b^2 (A b+3 a B) x^{15/2}+b^3 B x^{19/2}\right ) \, dx\\ &=\frac {2}{5} a^3 A x^{5/2}+\frac {2}{9} a^2 (3 A b+a B) x^{9/2}+\frac {6}{13} a b (A b+a B) x^{13/2}+\frac {2}{17} b^2 (A b+3 a B) x^{17/2}+\frac {2}{21} b^3 B x^{21/2}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 81, normalized size = 0.95 \begin {gather*} \frac {2 x^{5/2} \left (1547 a^3 \left (9 A+5 B x^2\right )+1785 a^2 b x^2 \left (13 A+9 B x^2\right )+945 a b^2 x^4 \left (17 A+13 B x^2\right )+195 b^3 x^6 \left (21 A+17 B x^2\right )\right )}{69615} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(5/2)*(1547*a^3*(9*A + 5*B*x^2) + 1785*a^2*b*x^2*(13*A + 9*B*x^2) + 945*a*b^2*x^4*(17*A + 13*B*x^2) + 195
*b^3*x^6*(21*A + 17*B*x^2)))/69615

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Maple [A]
time = 0.11, size = 76, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {2 b^{3} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{3}+3 B a \,b^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A a \,b^{2}+3 B \,a^{2} b \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\)	\(76\)
default	\(\frac {2 b^{3} B \,x^{\frac {21}{2}}}{21}+\frac {2 \left (A \,b^{3}+3 B a \,b^{2}\right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A a \,b^{2}+3 B \,a^{2} b \right ) x^{\frac {13}{2}}}{13}+\frac {2 \left (3 A \,a^{2} b +B \,a^{3}\right ) x^{\frac {9}{2}}}{9}+\frac {2 a^{3} A \,x^{\frac {5}{2}}}{5}\)	\(76\)
gosper	\(\frac {2 x^{\frac {5}{2}} \left (3315 B \,b^{3} x^{8}+4095 x^{6} A \,b^{3}+12285 x^{6} B a \,b^{2}+16065 A a \,b^{2} x^{4}+16065 x^{4} B \,a^{2} b +23205 x^{2} A \,a^{2} b +7735 B \,a^{3} x^{2}+13923 A \,a^{3}\right )}{69615}\)	\(80\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (3315 B \,b^{3} x^{8}+4095 x^{6} A \,b^{3}+12285 x^{6} B a \,b^{2}+16065 A a \,b^{2} x^{4}+16065 x^{4} B \,a^{2} b +23205 x^{2} A \,a^{2} b +7735 B \,a^{3} x^{2}+13923 A \,a^{3}\right )}{69615}\)	\(80\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (3315 B \,b^{3} x^{8}+4095 x^{6} A \,b^{3}+12285 x^{6} B a \,b^{2}+16065 A a \,b^{2} x^{4}+16065 x^{4} B \,a^{2} b +23205 x^{2} A \,a^{2} b +7735 B \,a^{3} x^{2}+13923 A \,a^{3}\right )}{69615}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*b^3*B*x^(21/2)+2/17*(A*b^3+3*B*a*b^2)*x^(17/2)+2/13*(3*A*a*b^2+3*B*a^2*b)*x^(13/2)+2/9*(3*A*a^2*b+B*a^3)*
x^(9/2)+2/5*a^3*A*x^(5/2)

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Maxima [A]
time = 0.29, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{21} \, B b^{3} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {17}{2}} + \frac {6}{13} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {13}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} + \frac {2}{9} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac {9}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*B*b^3*x^(21/2) + 2/17*(3*B*a*b^2 + A*b^3)*x^(17/2) + 6/13*(B*a^2*b + A*a*b^2)*x^(13/2) + 2/5*A*a^3*x^(5/2
) + 2/9*(B*a^3 + 3*A*a^2*b)*x^(9/2)

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Fricas [A]
time = 0.84, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{69615} \, {\left (3315 \, B b^{3} x^{10} + 4095 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{8} + 16065 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + 13923 \, A a^{3} x^{2} + 7735 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/69615*(3315*B*b^3*x^10 + 4095*(3*B*a*b^2 + A*b^3)*x^8 + 16065*(B*a^2*b + A*a*b^2)*x^6 + 13923*A*a^3*x^2 + 77
35*(B*a^3 + 3*A*a^2*b)*x^4)*sqrt(x)

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Sympy [A]
time = 0.75, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A a^{3} x^{\frac {5}{2}}}{5} + \frac {2 A a^{2} b x^{\frac {9}{2}}}{3} + \frac {6 A a b^{2} x^{\frac {13}{2}}}{13} + \frac {2 A b^{3} x^{\frac {17}{2}}}{17} + \frac {2 B a^{3} x^{\frac {9}{2}}}{9} + \frac {6 B a^{2} b x^{\frac {13}{2}}}{13} + \frac {6 B a b^{2} x^{\frac {17}{2}}}{17} + \frac {2 B b^{3} x^{\frac {21}{2}}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*a**3*x**(5/2)/5 + 2*A*a**2*b*x**(9/2)/3 + 6*A*a*b**2*x**(13/2)/13 + 2*A*b**3*x**(17/2)/17 + 2*B*a**3*x**(9
/2)/9 + 6*B*a**2*b*x**(13/2)/13 + 6*B*a*b**2*x**(17/2)/17 + 2*B*b**3*x**(21/2)/21

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Giac [A]
time = 0.62, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{21} \, B b^{3} x^{\frac {21}{2}} + \frac {6}{17} \, B a b^{2} x^{\frac {17}{2}} + \frac {2}{17} \, A b^{3} x^{\frac {17}{2}} + \frac {6}{13} \, B a^{2} b x^{\frac {13}{2}} + \frac {6}{13} \, A a b^{2} x^{\frac {13}{2}} + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} + \frac {2}{3} \, A a^{2} b x^{\frac {9}{2}} + \frac {2}{5} \, A a^{3} x^{\frac {5}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*B*b^3*x^(21/2) + 6/17*B*a*b^2*x^(17/2) + 2/17*A*b^3*x^(17/2) + 6/13*B*a^2*b*x^(13/2) + 6/13*A*a*b^2*x^(13
/2) + 2/9*B*a^3*x^(9/2) + 2/3*A*a^2*b*x^(9/2) + 2/5*A*a^3*x^(5/2)

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Mupad [B]
time = 0.02, size = 69, normalized size = 0.81 \begin {gather*} x^{9/2}\,\left (\frac {2\,B\,a^3}{9}+\frac {2\,A\,b\,a^2}{3}\right )+x^{17/2}\,\left (\frac {2\,A\,b^3}{17}+\frac {6\,B\,a\,b^2}{17}\right )+\frac {2\,A\,a^3\,x^{5/2}}{5}+\frac {2\,B\,b^3\,x^{21/2}}{21}+\frac {6\,a\,b\,x^{13/2}\,\left (A\,b+B\,a\right )}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(9/2)*((2*B*a^3)/9 + (2*A*a^2*b)/3) + x^(17/2)*((2*A*b^3)/17 + (6*B*a*b^2)/17) + (2*A*a^3*x^(5/2))/5 + (2*B*
b^3*x^(21/2))/21 + (6*a*b*x^(13/2)*(A*b + B*a))/13

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