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

Optimal. Leaf size=109 \[ -\frac {2 a^2 A}{\sqrt {x}}+\frac {2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt {x} (a B+2 A b)+\frac {2}{7} c x^{7/2} (A c+2 b B)+\frac {2}{9} B c^2 x^{9/2} \]

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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}{\sqrt {x}}+\frac {2}{5} x^{5/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{3} x^{3/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+2 a \sqrt {x} (a B+2 A b)+\frac {2}{7} c x^{7/2} (A c+2 b B)+\frac {2}{9} B c^2 x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.12, size = 97, normalized size = 0.89 \begin {gather*} \frac {-630 a^2 (A-B x)+84 a x (5 A (3 b+c x)+B x (5 b+3 c x))+2 x^2 \left (3 A \left (35 b^2+42 b c x+15 c^2 x^2\right )+B x \left (63 b^2+90 b c x+35 c^2 x^2\right )\right )}{315 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-630*a^2*(A - B*x) + 84*a*x*(5*A*(3*b + c*x) + B*x*(5*b + 3*c*x)) + 2*x^2*(3*A*(35*b^2 + 42*b*c*x + 15*c^2*x^
2) + B*x*(63*b^2 + 90*b*c*x + 35*c^2*x^2)))/(315*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.09, size = 105, normalized size = 0.96 \begin {gather*} \frac {2 \left (-315 a^2 A+315 a^2 B x+630 a A b x+210 a A c x^2+210 a b B x^2+126 a B c x^3+105 A b^2 x^2+126 A b c x^3+45 A c^2 x^4+63 b^2 B x^3+90 b B c x^4+35 B c^2 x^5\right )}{315 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-315*a^2*A + 630*a*A*b*x + 315*a^2*B*x + 105*A*b^2*x^2 + 210*a*b*B*x^2 + 210*a*A*c*x^2 + 63*b^2*B*x^3 + 12
6*A*b*c*x^3 + 126*a*B*c*x^3 + 90*b*B*c*x^4 + 45*A*c^2*x^4 + 35*B*c^2*x^5))/(315*Sqrt[x])

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fricas [A]  time = 0.44, size = 93, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (35 \, B c^{2} x^{5} + 45 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 63 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 315 \, A a^{2} + 105 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 315 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{315 \, \sqrt {x}} \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^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*c^2*x^5 + 45*(2*B*b*c + A*c^2)*x^4 + 63*(B*b^2 + 2*(B*a + A*b)*c)*x^3 - 315*A*a^2 + 105*(2*B*a*b +
A*b^2 + 2*A*a*c)*x^2 + 315*(B*a^2 + 2*A*a*b)*x)/sqrt(x)

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giac [A]  time = 0.16, size = 103, normalized size = 0.94 \begin {gather*} \frac {2}{9} \, B c^{2} x^{\frac {9}{2}} + \frac {4}{7} \, B b c x^{\frac {7}{2}} + \frac {2}{7} \, A c^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B b^{2} x^{\frac {5}{2}} + \frac {4}{5} \, B a c x^{\frac {5}{2}} + \frac {4}{5} \, A b c x^{\frac {5}{2}} + \frac {4}{3} \, B a b x^{\frac {3}{2}} + \frac {2}{3} \, A b^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a c x^{\frac {3}{2}} + 2 \, B a^{2} \sqrt {x} + 4 \, A a b \sqrt {x} - \frac {2 \, A a^{2}}{\sqrt {x}} \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^(3/2),x, algorithm="giac")

[Out]

2/9*B*c^2*x^(9/2) + 4/7*B*b*c*x^(7/2) + 2/7*A*c^2*x^(7/2) + 2/5*B*b^2*x^(5/2) + 4/5*B*a*c*x^(5/2) + 4/5*A*b*c*
x^(5/2) + 4/3*B*a*b*x^(3/2) + 2/3*A*b^2*x^(3/2) + 4/3*A*a*c*x^(3/2) + 2*B*a^2*sqrt(x) + 4*A*a*b*sqrt(x) - 2*A*
a^2/sqrt(x)

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maple [A]  time = 0.05, size = 102, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-35 B \,c^{2} x^{5}-45 A \,c^{2} x^{4}-90 x^{4} b B c -126 x^{3} A b c -126 B a c \,x^{3}-63 B \,b^{2} x^{3}-210 A a c \,x^{2}-105 A \,b^{2} x^{2}-210 B a b \,x^{2}-630 A a b x -315 B \,a^{2} x +315 A \,a^{2}\right )}{315 \sqrt {x}} \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^(3/2),x)

[Out]

-2/315*(-35*B*c^2*x^5-45*A*c^2*x^4-90*B*b*c*x^4-126*A*b*c*x^3-126*B*a*c*x^3-63*B*b^2*x^3-210*A*a*c*x^2-105*A*b
^2*x^2-210*B*a*b*x^2-630*A*a*b*x-315*B*a^2*x+315*A*a^2)/x^(1/2)

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maxima [A]  time = 0.56, size = 93, normalized size = 0.85 \begin {gather*} \frac {2}{9} \, B c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {5}{2}} - \frac {2 \, A a^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} \sqrt {x} \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^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 93, normalized size = 0.85 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^2+4\,A\,b\,a\right )+x^{7/2}\,\left (\frac {2\,A\,c^2}{7}+\frac {4\,B\,b\,c}{7}\right )+x^{3/2}\,\left (\frac {2\,A\,b^2}{3}+\frac {4\,B\,a\,b}{3}+\frac {4\,A\,a\,c}{3}\right )+x^{5/2}\,\left (\frac {2\,B\,b^2}{5}+\frac {4\,A\,c\,b}{5}+\frac {4\,B\,a\,c}{5}\right )-\frac {2\,A\,a^2}{\sqrt {x}}+\frac {2\,B\,c^2\,x^{9/2}}{9} \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^(3/2),x)

[Out]

x^(1/2)*(2*B*a^2 + 4*A*a*b) + x^(7/2)*((2*A*c^2)/7 + (4*B*b*c)/7) + x^(3/2)*((2*A*b^2)/3 + (4*A*a*c)/3 + (4*B*
a*b)/3) + x^(5/2)*((2*B*b^2)/5 + (4*A*b*c)/5 + (4*B*a*c)/5) - (2*A*a^2)/x^(1/2) + (2*B*c^2*x^(9/2))/9

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sympy [A]  time = 2.30, size = 156, normalized size = 1.43 \begin {gather*} - \frac {2 A a^{2}}{\sqrt {x}} + 4 A a b \sqrt {x} + \frac {4 A a c x^{\frac {3}{2}}}{3} + \frac {2 A b^{2} x^{\frac {3}{2}}}{3} + \frac {4 A b c x^{\frac {5}{2}}}{5} + \frac {2 A c^{2} x^{\frac {7}{2}}}{7} + 2 B a^{2} \sqrt {x} + \frac {4 B a b x^{\frac {3}{2}}}{3} + \frac {4 B a c x^{\frac {5}{2}}}{5} + \frac {2 B b^{2} x^{\frac {5}{2}}}{5} + \frac {4 B b c x^{\frac {7}{2}}}{7} + \frac {2 B c^{2} x^{\frac {9}{2}}}{9} \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**(3/2),x)

[Out]

-2*A*a**2/sqrt(x) + 4*A*a*b*sqrt(x) + 4*A*a*c*x**(3/2)/3 + 2*A*b**2*x**(3/2)/3 + 4*A*b*c*x**(5/2)/5 + 2*A*c**2
*x**(7/2)/7 + 2*B*a**2*sqrt(x) + 4*B*a*b*x**(3/2)/3 + 4*B*a*c*x**(5/2)/5 + 2*B*b**2*x**(5/2)/5 + 4*B*b*c*x**(7
/2)/7 + 2*B*c**2*x**(9/2)/9

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