3.5.10 \(\int \frac {(A+B x) (a+c x^2)^3}{x^{9/2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac {2 a^3 A}{7 x^{7/2}}-\frac {2 a^3 B}{5 x^{5/2}}-\frac {2 a^2 A c}{x^{3/2}}-\frac {6 a^2 B c}{\sqrt {x}}+6 a A c^2 \sqrt {x}+2 a B c^2 x^{3/2}+\frac {2}{5} A c^3 x^{5/2}+\frac {2}{7} B c^3 x^{7/2} \]

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Rubi [A]  time = 0.04, antiderivative size = 101, 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 c}{x^{3/2}}-\frac {2 a^3 A}{7 x^{7/2}}-\frac {6 a^2 B c}{\sqrt {x}}-\frac {2 a^3 B}{5 x^{5/2}}+6 a A c^2 \sqrt {x}+2 a B c^2 x^{3/2}+\frac {2}{5} A c^3 x^{5/2}+\frac {2}{7} B c^3 x^{7/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + c*x^2)^3)/x^(9/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 70, normalized size = 0.69 \begin {gather*} \frac {-2 a^3 (5 A+7 B x)-70 a^2 c x^2 (A+3 B x)+70 a c^2 x^4 (3 A+B x)+2 c^3 x^6 (7 A+5 B x)}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + c*x^2)^3)/x^(9/2),x]

[Out]

(70*a*c^2*x^4*(3*A + B*x) - 70*a^2*c*x^2*(A + 3*B*x) + 2*c^3*x^6*(7*A + 5*B*x) - 2*a^3*(5*A + 7*B*x))/(35*x^(7
/2))

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IntegrateAlgebraic [A]  time = 0.06, size = 81, normalized size = 0.80 \begin {gather*} \frac {2 \left (-5 a^3 A-7 a^3 B x-35 a^2 A c x^2-105 a^2 B c x^3+105 a A c^2 x^4+35 a B c^2 x^5+7 A c^3 x^6+5 B c^3 x^7\right )}{35 x^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a + c*x^2)^3)/x^(9/2),x]

[Out]

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

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fricas [A]  time = 0.42, size = 77, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (5 \, B c^{3} x^{7} + 7 \, A c^{3} x^{6} + 35 \, B a c^{2} x^{5} + 105 \, A a c^{2} x^{4} - 105 \, B a^{2} c x^{3} - 35 \, A a^{2} c x^{2} - 7 \, B a^{3} x - 5 \, A a^{3}\right )}}{35 \, 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)^3/x^(9/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.19, size = 78, normalized size = 0.77 \begin {gather*} \frac {2}{7} \, B c^{3} x^{\frac {7}{2}} + \frac {2}{5} \, A c^{3} x^{\frac {5}{2}} + 2 \, B a c^{2} x^{\frac {3}{2}} + 6 \, A a c^{2} \sqrt {x} - \frac {2 \, {\left (105 \, B a^{2} c x^{3} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 5 \, A a^{3}\right )}}{35 \, 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)^3/x^(9/2),x, algorithm="giac")

[Out]

2/7*B*c^3*x^(7/2) + 2/5*A*c^3*x^(5/2) + 2*B*a*c^2*x^(3/2) + 6*A*a*c^2*sqrt(x) - 2/35*(105*B*a^2*c*x^3 + 35*A*a
^2*c*x^2 + 7*B*a^3*x + 5*A*a^3)/x^(7/2)

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maple [A]  time = 0.05, size = 78, normalized size = 0.77 \begin {gather*} -\frac {2 \left (-5 B \,c^{3} x^{7}-7 A \,c^{3} x^{6}-35 B a \,c^{2} x^{5}-105 A a \,c^{2} x^{4}+105 B \,a^{2} c \,x^{3}+35 A \,a^{2} c \,x^{2}+7 B \,a^{3} x +5 A \,a^{3}\right )}{35 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)^3/x^(9/2),x)

[Out]

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

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maxima [A]  time = 0.52, size = 78, normalized size = 0.77 \begin {gather*} \frac {2}{7} \, B c^{3} x^{\frac {7}{2}} + \frac {2}{5} \, A c^{3} x^{\frac {5}{2}} + 2 \, B a c^{2} x^{\frac {3}{2}} + 6 \, A a c^{2} \sqrt {x} - \frac {2 \, {\left (105 \, B a^{2} c x^{3} + 35 \, A a^{2} c x^{2} + 7 \, B a^{3} x + 5 \, A a^{3}\right )}}{35 \, 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)^3/x^(9/2),x, algorithm="maxima")

[Out]

2/7*B*c^3*x^(7/2) + 2/5*A*c^3*x^(5/2) + 2*B*a*c^2*x^(3/2) + 6*A*a*c^2*sqrt(x) - 2/35*(105*B*a^2*c*x^3 + 35*A*a
^2*c*x^2 + 7*B*a^3*x + 5*A*a^3)/x^(7/2)

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mupad [B]  time = 0.03, size = 78, normalized size = 0.77 \begin {gather*} \frac {2\,A\,c^3\,x^{5/2}}{5}-\frac {\frac {2\,B\,a^3\,x}{5}+\frac {2\,A\,a^3}{7}+6\,B\,c\,a^2\,x^3+2\,A\,c\,a^2\,x^2}{x^{7/2}}+\frac {2\,B\,c^3\,x^{7/2}}{7}+6\,A\,a\,c^2\,\sqrt {x}+2\,B\,a\,c^2\,x^{3/2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + c*x^2)^3*(A + B*x))/x^(9/2),x)

[Out]

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

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sympy [A]  time = 8.61, size = 107, normalized size = 1.06 \begin {gather*} - \frac {2 A a^{3}}{7 x^{\frac {7}{2}}} - \frac {2 A a^{2} c}{x^{\frac {3}{2}}} + 6 A a c^{2} \sqrt {x} + \frac {2 A c^{3} x^{\frac {5}{2}}}{5} - \frac {2 B a^{3}}{5 x^{\frac {5}{2}}} - \frac {6 B a^{2} c}{\sqrt {x}} + 2 B a c^{2} x^{\frac {3}{2}} + \frac {2 B c^{3} x^{\frac {7}{2}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+a)**3/x**(9/2),x)

[Out]

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

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