3.409 \(\int \frac{(A+B x) (a+c x^2)^3}{x^{7/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0388654, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {766} \[ -\frac{6 a^2 A c}{\sqrt{x}}-\frac{2 a^3 A}{5 x^{5/2}}+6 a^2 B c \sqrt{x}-\frac{2 a^3 B}{3 x^{3/2}}+2 a A c^2 x^{3/2}+\frac{6}{5} a B c^2 x^{5/2}+\frac{2}{7} A c^3 x^{7/2}+\frac{2}{9} B c^3 x^{9/2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(7/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0268911, size = 72, normalized size = 0.7 \[ \frac{2 \left (945 a^2 c x^2 (B x-A)-21 a^3 (3 A+5 B x)+63 a c^2 x^4 (5 A+3 B x)+5 c^3 x^6 (9 A+7 B x)\right )}{315 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(7/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 78, normalized size = 0.8 \begin{align*} -{\frac{-70\,B{c}^{3}{x}^{7}-90\,A{c}^{3}{x}^{6}-378\,aB{c}^{2}{x}^{5}-630\,aA{c}^{2}{x}^{4}-1890\,{a}^{2}Bc{x}^{3}+1890\,{a}^{2}Ac{x}^{2}+210\,{a}^{3}Bx+126\,A{a}^{3}}{315}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+a)^3/x^(7/2),x)

[Out]

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

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Maxima [A]  time = 1.03166, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a c^{2} x^{\frac{5}{2}} + 2 \, A a c^{2} x^{\frac{3}{2}} + 6 \, B a^{2} c \sqrt{x} - \frac{2 \,{\left (45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 3 \, A a^{3}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(7/2),x, algorithm="maxima")

[Out]

2/9*B*c^3*x^(9/2) + 2/7*A*c^3*x^(7/2) + 6/5*B*a*c^2*x^(5/2) + 2*A*a*c^2*x^(3/2) + 6*B*a^2*c*sqrt(x) - 2/15*(45
*A*a^2*c*x^2 + 5*B*a^3*x + 3*A*a^3)/x^(5/2)

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Fricas [A]  time = 1.30698, size = 192, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (35 \, B c^{3} x^{7} + 45 \, A c^{3} x^{6} + 189 \, B a c^{2} x^{5} + 315 \, A a c^{2} x^{4} + 945 \, B a^{2} c x^{3} - 945 \, A a^{2} c x^{2} - 105 \, B a^{3} x - 63 \, A a^{3}\right )}}{315 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^3*x^7 + 45*A*c^3*x^6 + 189*B*a*c^2*x^5 + 315*A*a*c^2*x^4 + 945*B*a^2*c*x^3 - 945*A*a^2*c*x^2 - 1
05*B*a^3*x - 63*A*a^3)/x^(5/2)

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Sympy [A]  time = 8.00206, size = 109, normalized size = 1.06 \begin{align*} - \frac{2 A a^{3}}{5 x^{\frac{5}{2}}} - \frac{6 A a^{2} c}{\sqrt{x}} + 2 A a c^{2} x^{\frac{3}{2}} + \frac{2 A c^{3} x^{\frac{7}{2}}}{7} - \frac{2 B a^{3}}{3 x^{\frac{3}{2}}} + 6 B a^{2} c \sqrt{x} + \frac{6 B a c^{2} x^{\frac{5}{2}}}{5} + \frac{2 B c^{3} x^{\frac{9}{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(7/2),x)

[Out]

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

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Giac [A]  time = 1.1455, size = 105, normalized size = 1.02 \begin{align*} \frac{2}{9} \, B c^{3} x^{\frac{9}{2}} + \frac{2}{7} \, A c^{3} x^{\frac{7}{2}} + \frac{6}{5} \, B a c^{2} x^{\frac{5}{2}} + 2 \, A a c^{2} x^{\frac{3}{2}} + 6 \, B a^{2} c \sqrt{x} - \frac{2 \,{\left (45 \, A a^{2} c x^{2} + 5 \, B a^{3} x + 3 \, A a^{3}\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+a)^3/x^(7/2),x, algorithm="giac")

[Out]

2/9*B*c^3*x^(9/2) + 2/7*A*c^3*x^(7/2) + 6/5*B*a*c^2*x^(5/2) + 2*A*a*c^2*x^(3/2) + 6*B*a^2*c*sqrt(x) - 2/15*(45
*A*a^2*c*x^2 + 5*B*a^3*x + 3*A*a^3)/x^(5/2)