3.2.47 \(\int \frac {(A+B x) (b x+c x^2)}{x^{7/2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \]

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Rubi [A]  time = 0.02, antiderivative size = 35, 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 = {765} \begin {gather*} -\frac {2 (A c+b B)}{\sqrt {x}}-\frac {2 A b}{3 x^{3/2}}+2 B c \sqrt {x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*b)/(3*x^(3/2)) - (2*(b*B + A*c))/Sqrt[x] + 2*B*c*Sqrt[x]

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} -\frac {2 (A (b+3 c x)+3 B x (b-c x))}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(3*B*x*(b - c*x) + A*(b + 3*c*x)))/(3*x^(3/2))

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IntegrateAlgebraic [A]  time = 0.03, size = 31, normalized size = 0.89 \begin {gather*} \frac {2 \left (-A b-3 A c x-3 b B x+3 B c x^2\right )}{3 x^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(b*x + c*x^2))/x^(7/2),x]

[Out]

(2*(-(A*b) - 3*b*B*x - 3*A*c*x + 3*B*c*x^2))/(3*x^(3/2))

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fricas [A]  time = 0.40, size = 27, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (3 \, B c x^{2} - A b - 3 \, {\left (B b + A c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*B*c*x^2 - A*b - 3*(B*b + A*c)*x)/x^(3/2)

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giac [A]  time = 0.15, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (3 \, B b x + 3 \, A c x + A b\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*c*sqrt(x) - 2/3*(3*B*b*x + 3*A*c*x + A*b)/x^(3/2)

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maple [A]  time = 0.05, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2 \left (-3 B c \,x^{2}+3 A c x +3 B b x +A b \right )}{3 x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3/x^(3/2)*(-3*B*c*x^2+3*A*c*x+3*B*b*x+A*b)

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maxima [A]  time = 0.59, size = 27, normalized size = 0.77 \begin {gather*} 2 \, B c \sqrt {x} - \frac {2 \, {\left (A b + 3 \, {\left (B b + A c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*B*c*sqrt(x) - 2/3*(A*b + 3*(B*b + A*c)*x)/x^(3/2)

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mupad [B]  time = 1.02, size = 27, normalized size = 0.77 \begin {gather*} -\frac {2\,A\,b+6\,A\,c\,x+6\,B\,b\,x-6\,B\,c\,x^2}{3\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*A*b + 6*A*c*x + 6*B*b*x - 6*B*c*x^2)/(3*x^(3/2))

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sympy [A]  time = 1.34, size = 41, normalized size = 1.17 \begin {gather*} - \frac {2 A b}{3 x^{\frac {3}{2}}} - \frac {2 A c}{\sqrt {x}} - \frac {2 B b}{\sqrt {x}} + 2 B c \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*A*b/(3*x**(3/2)) - 2*A*c/sqrt(x) - 2*B*b/sqrt(x) + 2*B*c*sqrt(x)

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