3.4.46 \(\int \frac {(a+b x^2)^3 (A+B x^2)}{x^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{3} a^2 x^{3/2} (a B+3 A b)+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{15} b^3 B x^{15/2} \]

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Rubi [A]  time = 0.04, 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 = {448} \begin {gather*} \frac {2}{3} a^2 x^{3/2} (a B+3 A b)-\frac {2 a^3 A}{\sqrt {x}}+\frac {2}{11} b^2 x^{11/2} (3 a B+A b)+\frac {6}{7} a b x^{7/2} (a B+A b)+\frac {2}{15} b^3 B x^{15/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2))/7 + (2*b^2*(A*b + 3*a*B)*x^
(11/2))/11 + (2*b^3*B*x^(15/2))/15

Rule 448

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

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Mathematica [A]  time = 0.02, size = 81, normalized size = 0.98 \begin {gather*} \frac {-770 a^3 \left (3 A-B x^2\right )+330 a^2 b x^2 \left (7 A+3 B x^2\right )+90 a b^2 x^4 \left (11 A+7 B x^2\right )+14 b^3 x^6 \left (15 A+11 B x^2\right )}{1155 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-770*a^3*(3*A - B*x^2) + 330*a^2*b*x^2*(7*A + 3*B*x^2) + 90*a*b^2*x^4*(11*A + 7*B*x^2) + 14*b^3*x^6*(15*A + 1
1*B*x^2))/(1155*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.05, size = 83, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1155 a^3 A+385 a^3 B x^2+1155 a^2 A b x^2+495 a^2 b B x^4+495 a A b^2 x^4+315 a b^2 B x^6+105 A b^3 x^6+77 b^3 B x^8\right )}{1155 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1155*a^3*A + 1155*a^2*A*b*x^2 + 385*a^3*B*x^2 + 495*a*A*b^2*x^4 + 495*a^2*b*B*x^4 + 105*A*b^3*x^6 + 315*a
*b^2*B*x^6 + 77*b^3*B*x^8))/(1155*Sqrt[x])

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fricas [A]  time = 1.03, size = 75, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (77 \, B b^{3} x^{8} + 105 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 495 \, {\left (B a^{2} b + A a b^{2}\right )} x^{4} - 1155 \, A a^{3} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{1155 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/1155*(77*B*b^3*x^8 + 105*(3*B*a*b^2 + A*b^3)*x^6 + 495*(B*a^2*b + A*a*b^2)*x^4 - 1155*A*a^3 + 385*(B*a^3 + 3
*A*a^2*b)*x^2)/sqrt(x)

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giac [A]  time = 0.30, size = 77, normalized size = 0.93 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {6}{11} \, B a b^{2} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{3} x^{\frac {11}{2}} + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} + \frac {2}{3} \, B a^{3} x^{\frac {3}{2}} + 2 \, A a^{2} b x^{\frac {3}{2}} - \frac {2 \, A a^{3}}{\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^(3/2),x, algorithm="giac")

[Out]

2/15*B*b^3*x^(15/2) + 6/11*B*a*b^2*x^(11/2) + 2/11*A*b^3*x^(11/2) + 6/7*B*a^2*b*x^(7/2) + 6/7*A*a*b^2*x^(7/2)
+ 2/3*B*a^3*x^(3/2) + 2*A*a^2*b*x^(3/2) - 2*A*a^3/sqrt(x)

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maple [A]  time = 0.01, size = 80, normalized size = 0.96 \begin {gather*} -\frac {2 \left (-77 B \,b^{3} x^{8}-105 x^{6} A \,b^{3}-315 B a \,b^{2} x^{6}-495 x^{4} A a \,b^{2}-495 x^{4} B \,a^{2} b -1155 A \,a^{2} b \,x^{2}-385 B \,a^{3} x^{2}+1155 A \,a^{3}\right )}{1155 \sqrt {x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1155*(-77*B*b^3*x^8-105*A*b^3*x^6-315*B*a*b^2*x^6-495*A*a*b^2*x^4-495*B*a^2*b*x^4-1155*A*a^2*b*x^2-385*B*a^
3*x^2+1155*A*a^3)/x^(1/2)

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maxima [A]  time = 0.96, size = 73, normalized size = 0.88 \begin {gather*} \frac {2}{15} \, B b^{3} x^{\frac {15}{2}} + \frac {2}{11} \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac {11}{2}} + \frac {6}{7} \, {\left (B a^{2} b + A a b^{2}\right )} x^{\frac {7}{2}} - \frac {2 \, A a^{3}}{\sqrt {x}} + \frac {2}{3} \, {\left (B a^{3} + 3 \, A a^{2} b\right )} 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^(3/2),x, algorithm="maxima")

[Out]

2/15*B*b^3*x^(15/2) + 2/11*(3*B*a*b^2 + A*b^3)*x^(11/2) + 6/7*(B*a^2*b + A*a*b^2)*x^(7/2) - 2*A*a^3/sqrt(x) +
2/3*(B*a^3 + 3*A*a^2*b)*x^(3/2)

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mupad [B]  time = 0.03, size = 69, normalized size = 0.83 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^3}{3}+2\,A\,b\,a^2\right )+x^{11/2}\,\left (\frac {2\,A\,b^3}{11}+\frac {6\,B\,a\,b^2}{11}\right )-\frac {2\,A\,a^3}{\sqrt {x}}+\frac {2\,B\,b^3\,x^{15/2}}{15}+\frac {6\,a\,b\,x^{7/2}\,\left (A\,b+B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(3/2)*((2*B*a^3)/3 + 2*A*a^2*b) + x^(11/2)*((2*A*b^3)/11 + (6*B*a*b^2)/11) - (2*A*a^3)/x^(1/2) + (2*B*b^3*x^
(15/2))/15 + (6*a*b*x^(7/2)*(A*b + B*a))/7

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sympy [A]  time = 5.33, size = 110, normalized size = 1.33 \begin {gather*} - \frac {2 A a^{3}}{\sqrt {x}} + 2 A a^{2} b x^{\frac {3}{2}} + \frac {6 A a b^{2} x^{\frac {7}{2}}}{7} + \frac {2 A b^{3} x^{\frac {11}{2}}}{11} + \frac {2 B a^{3} x^{\frac {3}{2}}}{3} + \frac {6 B a^{2} b x^{\frac {7}{2}}}{7} + \frac {6 B a b^{2} x^{\frac {11}{2}}}{11} + \frac {2 B b^{3} x^{\frac {15}{2}}}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*a**3/sqrt(x) + 2*A*a**2*b*x**(3/2) + 6*A*a*b**2*x**(7/2)/7 + 2*A*b**3*x**(11/2)/11 + 2*B*a**3*x**(3/2)/3
+ 6*B*a**2*b*x**(7/2)/7 + 6*B*a*b**2*x**(11/2)/11 + 2*B*b**3*x**(15/2)/15

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