3.5.11 \(\int \frac {(A+B x) (a+c x^2)^3}{x^{11/2}} \, dx\)

Optimal. Leaf size=103 \[ -\frac {2 a^3 A}{9 x^{9/2}}-\frac {2 a^3 B}{7 x^{7/2}}-\frac {6 a^2 A c}{5 x^{5/2}}-\frac {2 a^2 B c}{x^{3/2}}-\frac {6 a A c^2}{\sqrt {x}}+6 a B c^2 \sqrt {x}+\frac {2}{3} A c^3 x^{3/2}+\frac {2}{5} B c^3 x^{5/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} -\frac {6 a^2 A c}{5 x^{5/2}}-\frac {2 a^3 A}{9 x^{9/2}}-\frac {2 a^2 B c}{x^{3/2}}-\frac {2 a^3 B}{7 x^{7/2}}-\frac {6 a A c^2}{\sqrt {x}}+6 a B c^2 \sqrt {x}+\frac {2}{3} A c^3 x^{3/2}+\frac {2}{5} B c^3 x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*a^3*A)/(9*x^(9/2)) - (2*a^3*B)/(7*x^(7/2)) - (6*a^2*A*c)/(5*x^(5/2)) - (2*a^2*B*c)/x^(3/2) - (6*a*A*c^2)/S
qrt[x] + 6*a*B*c^2*Sqrt[x] + (2*A*c^3*x^(3/2))/3 + (2*B*c^3*x^(5/2))/5

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 71, normalized size = 0.69 \begin {gather*} -\frac {2 \left (5 a^3 (7 A+9 B x)+63 a^2 c x^2 (3 A+5 B x)+945 a c^2 x^4 (A-B x)-21 c^3 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(11/2),x]

[Out]

(-2*(945*a*c^2*x^4*(A - B*x) - 21*c^3*x^6*(5*A + 3*B*x) + 63*a^2*c*x^2*(3*A + 5*B*x) + 5*a^3*(7*A + 9*B*x)))/(
315*x^(9/2))

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IntegrateAlgebraic [A]  time = 0.05, size = 81, normalized size = 0.79 \begin {gather*} \frac {2 \left (-35 a^3 A-45 a^3 B x-189 a^2 A c x^2-315 a^2 B c x^3-945 a A c^2 x^4+945 a B c^2 x^5+105 A c^3 x^6+63 B c^3 x^7\right )}{315 x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/x^(11/2),x]

[Out]

(2*(-35*a^3*A - 45*a^3*B*x - 189*a^2*A*c*x^2 - 315*a^2*B*c*x^3 - 945*a*A*c^2*x^4 + 945*a*B*c^2*x^5 + 105*A*c^3
*x^6 + 63*B*c^3*x^7))/(315*x^(9/2))

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fricas [A]  time = 0.40, size = 77, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (63 \, B c^{3} x^{7} + 105 \, A c^{3} x^{6} + 945 \, B a c^{2} x^{5} - 945 \, A a c^{2} x^{4} - 315 \, B a^{2} c x^{3} - 189 \, A a^{2} c x^{2} - 45 \, B a^{3} x - 35 \, A a^{3}\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="fricas")

[Out]

2/315*(63*B*c^3*x^7 + 105*A*c^3*x^6 + 945*B*a*c^2*x^5 - 945*A*a*c^2*x^4 - 315*B*a^2*c*x^3 - 189*A*a^2*c*x^2 -
45*B*a^3*x - 35*A*a^3)/x^(9/2)

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giac [A]  time = 0.19, size = 78, normalized size = 0.76 \begin {gather*} \frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + \frac {2}{3} \, A c^{3} x^{\frac {3}{2}} + 6 \, B a c^{2} \sqrt {x} - \frac {2 \, {\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="giac")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*A*c^3*x^(3/2) + 6*B*a*c^2*sqrt(x) - 2/315*(945*A*a*c^2*x^4 + 315*B*a^2*c*x^3 + 189*A*a
^2*c*x^2 + 45*B*a^3*x + 35*A*a^3)/x^(9/2)

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maple [A]  time = 0.05, size = 78, normalized size = 0.76 \begin {gather*} -\frac {2 \left (-63 B \,c^{3} x^{7}-105 A \,c^{3} x^{6}-945 B a \,c^{2} x^{5}+945 A a \,c^{2} x^{4}+315 B \,a^{2} c \,x^{3}+189 A \,a^{2} c \,x^{2}+45 B \,a^{3} x +35 A \,a^{3}\right )}{315 x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/x^(11/2),x)

[Out]

-2/315*(-63*B*c^3*x^7-105*A*c^3*x^6-945*B*a*c^2*x^5+945*A*a*c^2*x^4+315*B*a^2*c*x^3+189*A*a^2*c*x^2+45*B*a^3*x
+35*A*a^3)/x^(9/2)

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maxima [A]  time = 0.52, size = 78, normalized size = 0.76 \begin {gather*} \frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + \frac {2}{3} \, A c^{3} x^{\frac {3}{2}} + 6 \, B a c^{2} \sqrt {x} - \frac {2 \, {\left (945 \, A a c^{2} x^{4} + 315 \, B a^{2} c x^{3} + 189 \, A a^{2} c x^{2} + 45 \, B a^{3} x + 35 \, A a^{3}\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(11/2),x, algorithm="maxima")

[Out]

2/5*B*c^3*x^(5/2) + 2/3*A*c^3*x^(3/2) + 6*B*a*c^2*sqrt(x) - 2/315*(945*A*a*c^2*x^4 + 315*B*a^2*c*x^3 + 189*A*a
^2*c*x^2 + 45*B*a^3*x + 35*A*a^3)/x^(9/2)

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mupad [B]  time = 0.06, size = 78, normalized size = 0.76 \begin {gather*} \frac {2\,A\,c^3\,x^{3/2}}{3}-\frac {\frac {2\,B\,a^3\,x}{7}+\frac {2\,A\,a^3}{9}+2\,B\,a^2\,c\,x^3+\frac {6\,A\,a^2\,c\,x^2}{5}+6\,A\,a\,c^2\,x^4}{x^{9/2}}+\frac {2\,B\,c^3\,x^{5/2}}{5}+6\,B\,a\,c^2\,\sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/x^(11/2),x)

[Out]

(2*A*c^3*x^(3/2))/3 - ((2*A*a^3)/9 + (2*B*a^3*x)/7 + (6*A*a^2*c*x^2)/5 + 6*A*a*c^2*x^4 + 2*B*a^2*c*x^3)/x^(9/2
) + (2*B*c^3*x^(5/2))/5 + 6*B*a*c^2*x^(1/2)

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sympy [A]  time = 11.24, size = 109, normalized size = 1.06 \begin {gather*} - \frac {2 A a^{3}}{9 x^{\frac {9}{2}}} - \frac {6 A a^{2} c}{5 x^{\frac {5}{2}}} - \frac {6 A a c^{2}}{\sqrt {x}} + \frac {2 A c^{3} x^{\frac {3}{2}}}{3} - \frac {2 B a^{3}}{7 x^{\frac {7}{2}}} - \frac {2 B a^{2} c}{x^{\frac {3}{2}}} + 6 B a c^{2} \sqrt {x} + \frac {2 B c^{3} x^{\frac {5}{2}}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(11/2),x)

[Out]

-2*A*a**3/(9*x**(9/2)) - 6*A*a**2*c/(5*x**(5/2)) - 6*A*a*c**2/sqrt(x) + 2*A*c**3*x**(3/2)/3 - 2*B*a**3/(7*x**(
7/2)) - 2*B*a**2*c/x**(3/2) + 6*B*a*c**2*sqrt(x) + 2*B*c**3*x**(5/2)/5

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