3.5.1 \(\int \frac {(A+B x) (a+c x^2)^2}{x^{9/2}} \, dx\)

Optimal. Leaf size=73 \[ -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a^2 B}{5 x^{5/2}}-\frac {4 a A c}{3 x^{3/2}}-\frac {4 a B c}{\sqrt {x}}+2 A c^2 \sqrt {x}+\frac {2}{3} B c^2 x^{3/2} \]

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Rubi [A]  time = 0.02, antiderivative size = 73, 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 {2 a^2 A}{7 x^{7/2}}-\frac {2 a^2 B}{5 x^{5/2}}-\frac {4 a A c}{3 x^{3/2}}-\frac {4 a B c}{\sqrt {x}}+2 A c^2 \sqrt {x}+\frac {2}{3} B c^2 x^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^2)/x^(9/2),x]

[Out]

(-2*a^2*A)/(7*x^(7/2)) - (2*a^2*B)/(5*x^(5/2)) - (4*a*A*c)/(3*x^(3/2)) - (4*a*B*c)/Sqrt[x] + 2*A*c^2*Sqrt[x] +
(2*B*c^2*x^(3/2))/3

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 )^2}{x^{9/2}} \, dx &=\int \left (\frac {a^2 A}{x^{9/2}}+\frac {a^2 B}{x^{7/2}}+\frac {2 a A c}{x^{5/2}}+\frac {2 a B c}{x^{3/2}}+\frac {A c^2}{\sqrt {x}}+B c^2 \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a^2 B}{5 x^{5/2}}-\frac {4 a A c}{3 x^{3/2}}-\frac {4 a B c}{\sqrt {x}}+2 A c^2 \sqrt {x}+\frac {2}{3} B c^2 x^{3/2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 51, normalized size = 0.70 \begin {gather*} \frac {-6 a^2 (5 A+7 B x)-140 a c x^2 (A+3 B x)+70 c^2 x^4 (3 A+B x)}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^2)/x^(9/2),x]

[Out]

(70*c^2*x^4*(3*A + B*x) - 140*a*c*x^2*(A + 3*B*x) - 6*a^2*(5*A + 7*B*x))/(105*x^(7/2))

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IntegrateAlgebraic [A]  time = 0.05, size = 57, normalized size = 0.78 \begin {gather*} \frac {2 \left (-15 a^2 A-21 a^2 B x-70 a A c x^2-210 a B c x^3+105 A c^2 x^4+35 B c^2 x^5\right )}{105 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^2)/x^(9/2),x]

[Out]

(2*(-15*a^2*A - 21*a^2*B*x - 70*a*A*c*x^2 - 210*a*B*c*x^3 + 105*A*c^2*x^4 + 35*B*c^2*x^5))/(105*x^(7/2))

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fricas [A]  time = 0.41, size = 53, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (35 \, B c^{2} x^{5} + 105 \, A c^{2} x^{4} - 210 \, B a c x^{3} - 70 \, A a c x^{2} - 21 \, B a^{2} x - 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(9/2),x, algorithm="fricas")

[Out]

2/105*(35*B*c^2*x^5 + 105*A*c^2*x^4 - 210*B*a*c*x^3 - 70*A*a*c*x^2 - 21*B*a^2*x - 15*A*a^2)/x^(7/2)

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giac [A]  time = 0.15, size = 54, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (210 \, B a c x^{3} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(9/2),x, algorithm="giac")

[Out]

2/3*B*c^2*x^(3/2) + 2*A*c^2*sqrt(x) - 2/105*(210*B*a*c*x^3 + 70*A*a*c*x^2 + 21*B*a^2*x + 15*A*a^2)/x^(7/2)

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maple [A]  time = 0.05, size = 54, normalized size = 0.74 \begin {gather*} -\frac {2 \left (-35 B \,c^{2} x^{5}-105 A \,c^{2} x^{4}+210 B a c \,x^{3}+70 A a c \,x^{2}+21 B \,a^{2} x +15 A \,a^{2}\right )}{105 x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^2/x^(9/2),x)

[Out]

-2/105*(-35*B*c^2*x^5-105*A*c^2*x^4+210*B*a*c*x^3+70*A*a*c*x^2+21*B*a^2*x+15*A*a^2)/x^(7/2)

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maxima [A]  time = 0.48, size = 54, normalized size = 0.74 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (210 \, B a c x^{3} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^2/x^(9/2),x, algorithm="maxima")

[Out]

2/3*B*c^2*x^(3/2) + 2*A*c^2*sqrt(x) - 2/105*(210*B*a*c*x^3 + 70*A*a*c*x^2 + 21*B*a^2*x + 15*A*a^2)/x^(7/2)

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mupad [B]  time = 0.05, size = 53, normalized size = 0.73 \begin {gather*} -\frac {42\,B\,a^2\,x+30\,A\,a^2+420\,B\,a\,c\,x^3+140\,A\,a\,c\,x^2-70\,B\,c^2\,x^5-210\,A\,c^2\,x^4}{105\,x^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^2*(A + B*x))/x^(9/2),x)

[Out]

-(30*A*a^2 - 210*A*c^2*x^4 - 70*B*c^2*x^5 + 42*B*a^2*x + 140*A*a*c*x^2 + 420*B*a*c*x^3)/(105*x^(7/2))

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sympy [A]  time = 4.02, size = 76, normalized size = 1.04 \begin {gather*} - \frac {2 A a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 A a c}{3 x^{\frac {3}{2}}} + 2 A c^{2} \sqrt {x} - \frac {2 B a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 B a c}{\sqrt {x}} + \frac {2 B c^{2} x^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**2/x**(9/2),x)

[Out]

-2*A*a**2/(7*x**(7/2)) - 4*A*a*c/(3*x**(3/2)) + 2*A*c**2*sqrt(x) - 2*B*a**2/(5*x**(5/2)) - 4*B*a*c/sqrt(x) + 2
*B*c**2*x**(3/2)/3

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