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

Optimal. Leaf size=109 \[ -\frac {2 a^2 A}{5 x^{5/2}}+2 \sqrt {x} \left (2 a B c+2 A b c+b^2 B\right )-\frac {2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{\sqrt {x}}-\frac {2 a (a B+2 A b)}{3 x^{3/2}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+\frac {2}{5} B c^2 x^{5/2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {765} \begin {gather*} -\frac {2 a^2 A}{5 x^{5/2}}+2 \sqrt {x} \left (2 a B c+2 A b c+b^2 B\right )-\frac {2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{\sqrt {x}}-\frac {2 a (a B+2 A b)}{3 x^{3/2}}+\frac {2}{3} c x^{3/2} (A c+2 b B)+\frac {2}{5} B c^2 x^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 95, normalized size = 0.87 \begin {gather*} \frac {-2 a^2 (3 A+5 B x)-20 a x (A (b+3 c x)+3 B x (b-c x))+2 x^2 \left (5 A \left (-3 b^2+6 b c x+c^2 x^2\right )+B x \left (15 b^2+10 b c x+3 c^2 x^2\right )\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^2*(3*A + 5*B*x) - 20*a*x*(3*B*x*(b - c*x) + A*(b + 3*c*x)) + 2*x^2*(5*A*(-3*b^2 + 6*b*c*x + c^2*x^2) + B
*x*(15*b^2 + 10*b*c*x + 3*c^2*x^2)))/(15*x^(5/2))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.09, size = 105, normalized size = 0.96 \begin {gather*} \frac {2 \left (-3 a^2 A-5 a^2 B x-10 a A b x-30 a A c x^2-30 a b B x^2+30 a B c x^3-15 A b^2 x^2+30 A b c x^3+5 A c^2 x^4+15 b^2 B x^3+10 b B c x^4+3 B c^2 x^5\right )}{15 x^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-3*a^2*A - 10*a*A*b*x - 5*a^2*B*x - 15*A*b^2*x^2 - 30*a*b*B*x^2 - 30*a*A*c*x^2 + 15*b^2*B*x^3 + 30*A*b*c*x
^3 + 30*a*B*c*x^3 + 10*b*B*c*x^4 + 5*A*c^2*x^4 + 3*B*c^2*x^5))/(15*x^(5/2))

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 93, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (3 \, B c^{2} x^{5} + 5 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 15 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 3 \, A a^{2} - 15 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*B*c^2*x^5 + 5*(2*B*b*c + A*c^2)*x^4 + 15*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 3*A*a^2 - 15*(2*B*a*b + A*b^2
+ 2*A*a*c)*x^2 - 5*(B*a^2 + 2*A*a*b)*x)/x^(5/2)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 102, normalized size = 0.94 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {4}{3} \, B b c x^{\frac {3}{2}} + \frac {2}{3} \, A c^{2} x^{\frac {3}{2}} + 2 \, B b^{2} \sqrt {x} + 4 \, B a c \sqrt {x} + 4 \, A b c \sqrt {x} - \frac {2 \, {\left (30 \, B a b x^{2} + 15 \, A b^{2} x^{2} + 30 \, A a c x^{2} + 5 \, B a^{2} x + 10 \, A a b x + 3 \, A a^{2}\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*B*c^2*x^(5/2) + 4/3*B*b*c*x^(3/2) + 2/3*A*c^2*x^(3/2) + 2*B*b^2*sqrt(x) + 4*B*a*c*sqrt(x) + 4*A*b*c*sqrt(x
) - 2/15*(30*B*a*b*x^2 + 15*A*b^2*x^2 + 30*A*a*c*x^2 + 5*B*a^2*x + 10*A*a*b*x + 3*A*a^2)/x^(5/2)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 102, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-3 B \,c^{2} x^{5}-5 A \,c^{2} x^{4}-10 x^{4} b B c -30 x^{3} A b c -30 B a c \,x^{3}-15 B \,b^{2} x^{3}+30 A a c \,x^{2}+15 A \,b^{2} x^{2}+30 B a b \,x^{2}+10 A a b x +5 B \,a^{2} x +3 A \,a^{2}\right )}{15 x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(-3*B*c^2*x^5-5*A*c^2*x^4-10*B*b*c*x^4-30*A*b*c*x^3-30*B*a*c*x^3-15*B*b^2*x^3+30*A*a*c*x^2+15*A*b^2*x^2+
30*B*a*b*x^2+10*A*a*b*x+5*B*a^2*x+3*A*a^2)/x^(5/2)

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 94, normalized size = 0.86 \begin {gather*} \frac {2}{5} \, B c^{2} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} \sqrt {x} - \frac {2 \, {\left (3 \, A a^{2} + 15 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{15 \, x^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*B*c^2*x^(5/2) + 2/3*(2*B*b*c + A*c^2)*x^(3/2) + 2*(B*b^2 + 2*(B*a + A*b)*c)*sqrt(x) - 2/15*(3*A*a^2 + 15*(
2*B*a*b + A*b^2 + 2*A*a*c)*x^2 + 5*(B*a^2 + 2*A*a*b)*x)/x^(5/2)

________________________________________________________________________________________

mupad [B]  time = 0.06, size = 94, normalized size = 0.86 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,c^2}{3}+\frac {4\,B\,b\,c}{3}\right )-\frac {\frac {2\,A\,a^2}{5}+x^2\,\left (2\,A\,b^2+4\,B\,a\,b+4\,A\,a\,c\right )+x\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )}{x^{5/2}}+\sqrt {x}\,\left (2\,B\,b^2+4\,A\,c\,b+4\,B\,a\,c\right )+\frac {2\,B\,c^2\,x^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 3.94, size = 151, normalized size = 1.39 \begin {gather*} - \frac {2 A a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 A a b}{3 x^{\frac {3}{2}}} - \frac {4 A a c}{\sqrt {x}} - \frac {2 A b^{2}}{\sqrt {x}} + 4 A b c \sqrt {x} + \frac {2 A c^{2} x^{\frac {3}{2}}}{3} - \frac {2 B a^{2}}{3 x^{\frac {3}{2}}} - \frac {4 B a b}{\sqrt {x}} + 4 B a c \sqrt {x} + 2 B b^{2} \sqrt {x} + \frac {4 B b c x^{\frac {3}{2}}}{3} + \frac {2 B c^{2} 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+b*x+a)**2/x**(7/2),x)

[Out]

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

________________________________________________________________________________________