3.10.19 \(\int \frac {(A+B x) (a+b x+c x^2)^2}{\sqrt {x}} \, dx\)

Optimal. Leaf size=111 \[ 2 a^2 A \sqrt {x}+\frac {2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{9} c x^{9/2} (A c+2 b B)+\frac {2}{11} B c^2 x^{11/2} \]

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Rubi [A]  time = 0.06, antiderivative size = 111, 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*} 2 a^2 A \sqrt {x}+\frac {2}{7} x^{7/2} \left (2 a B c+2 A b c+b^2 B\right )+\frac {2}{5} x^{5/2} \left (A \left (2 a c+b^2\right )+2 a b B\right )+\frac {2}{3} a x^{3/2} (a B+2 A b)+\frac {2}{9} c x^{9/2} (A c+2 b B)+\frac {2}{11} B c^2 x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^2*A*Sqrt[x] + (2*a*(2*A*b + a*B)*x^(3/2))/3 + (2*(2*a*b*B + A*(b^2 + 2*a*c))*x^(5/2))/5 + (2*(b^2*B + 2*A*
b*c + 2*a*B*c)*x^(7/2))/7 + (2*c*(2*b*B + A*c)*x^(9/2))/9 + (2*B*c^2*x^(11/2))/11

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}{\sqrt {x}} \, dx &=\int \left (\frac {a^2 A}{\sqrt {x}}+a (2 A b+a B) \sqrt {x}+\left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{3/2}+\left (b^2 B+2 A b c+2 a B c\right ) x^{5/2}+c (2 b B+A c) x^{7/2}+B c^2 x^{9/2}\right ) \, dx\\ &=2 a^2 A \sqrt {x}+\frac {2}{3} a (2 A b+a B) x^{3/2}+\frac {2}{5} \left (2 a b B+A \left (b^2+2 a c\right )\right ) x^{5/2}+\frac {2}{7} \left (b^2 B+2 A b c+2 a B c\right ) x^{7/2}+\frac {2}{9} c (2 b B+A c) x^{9/2}+\frac {2}{11} B c^2 x^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 100, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {x} \left (1155 a^2 (3 A+B x)+66 a x (7 A (5 b+3 c x)+3 B x (7 b+5 c x))+x^2 \left (11 A \left (63 b^2+90 b c x+35 c^2 x^2\right )+5 B x \left (99 b^2+154 b c x+63 c^2 x^2\right )\right )\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(1155*a^2*(3*A + B*x) + 66*a*x*(7*A*(5*b + 3*c*x) + 3*B*x*(7*b + 5*c*x)) + x^2*(11*A*(63*b^2 + 90*b
*c*x + 35*c^2*x^2) + 5*B*x*(99*b^2 + 154*b*c*x + 63*c^2*x^2))))/3465

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IntegrateAlgebraic [A]  time = 0.06, size = 131, normalized size = 1.18 \begin {gather*} \frac {2 \left (3465 a^2 A \sqrt {x}+1155 a^2 B x^{3/2}+2310 a A b x^{3/2}+1386 a A c x^{5/2}+1386 a b B x^{5/2}+990 a B c x^{7/2}+693 A b^2 x^{5/2}+990 A b c x^{7/2}+385 A c^2 x^{9/2}+495 b^2 B x^{7/2}+770 b B c x^{9/2}+315 B c^2 x^{11/2}\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3465*a^2*A*Sqrt[x] + 2310*a*A*b*x^(3/2) + 1155*a^2*B*x^(3/2) + 693*A*b^2*x^(5/2) + 1386*a*b*B*x^(5/2) + 13
86*a*A*c*x^(5/2) + 495*b^2*B*x^(7/2) + 990*A*b*c*x^(7/2) + 990*a*B*c*x^(7/2) + 770*b*B*c*x^(9/2) + 385*A*c^2*x
^(9/2) + 315*B*c^2*x^(11/2)))/3465

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fricas [A]  time = 0.41, size = 93, normalized size = 0.84 \begin {gather*} \frac {2}{3465} \, {\left (315 \, B c^{2} x^{5} + 385 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 495 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 3465 \, A a^{2} + 693 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 1155 \, {\left (B a^{2} + 2 \, A a b\right )} x\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^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*c^2*x^5 + 385*(2*B*b*c + A*c^2)*x^4 + 495*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 3465*A*a^2 + 693*(2*B*
a*b + A*b^2 + 2*A*a*c)*x^2 + 1155*(B*a^2 + 2*A*a*b)*x)*sqrt(x)

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giac [A]  time = 0.16, size = 103, normalized size = 0.93 \begin {gather*} \frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {4}{9} \, B b c x^{\frac {9}{2}} + \frac {2}{9} \, A c^{2} x^{\frac {9}{2}} + \frac {2}{7} \, B b^{2} x^{\frac {7}{2}} + \frac {4}{7} \, B a c x^{\frac {7}{2}} + \frac {4}{7} \, A b c x^{\frac {7}{2}} + \frac {4}{5} \, B a b x^{\frac {5}{2}} + \frac {2}{5} \, A b^{2} x^{\frac {5}{2}} + \frac {4}{5} \, A a c x^{\frac {5}{2}} + \frac {2}{3} \, B a^{2} x^{\frac {3}{2}} + \frac {4}{3} \, A a b x^{\frac {3}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

2/11*B*c^2*x^(11/2) + 4/9*B*b*c*x^(9/2) + 2/9*A*c^2*x^(9/2) + 2/7*B*b^2*x^(7/2) + 4/7*B*a*c*x^(7/2) + 4/7*A*b*
c*x^(7/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 4/5*A*a*c*x^(5/2) + 2/3*B*a^2*x^(3/2) + 4/3*A*a*b*x^(3/2)
+ 2*A*a^2*sqrt(x)

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maple [A]  time = 0.05, size = 102, normalized size = 0.92 \begin {gather*} \frac {2 \left (315 B \,c^{2} x^{5}+385 A \,c^{2} x^{4}+770 x^{4} b B c +990 x^{3} A b c +990 B a c \,x^{3}+495 B \,b^{2} x^{3}+1386 A a c \,x^{2}+693 A \,b^{2} x^{2}+1386 B a b \,x^{2}+2310 A a b x +1155 B \,a^{2} x +3465 A \,a^{2}\right ) \sqrt {x}}{3465} \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^(1/2),x)

[Out]

2/3465*x^(1/2)*(315*B*c^2*x^5+385*A*c^2*x^4+770*B*b*c*x^4+990*A*b*c*x^3+990*B*a*c*x^3+495*B*b^2*x^3+1386*A*a*c
*x^2+693*A*b^2*x^2+1386*B*a*b*x^2+2310*A*a*b*x+1155*B*a^2*x+3465*A*a^2)

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maxima [A]  time = 0.53, size = 93, normalized size = 0.84 \begin {gather*} \frac {2}{11} \, B c^{2} x^{\frac {11}{2}} + \frac {2}{9} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {9}{2}} + \frac {2}{7} \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{\frac {7}{2}} + 2 \, A a^{2} \sqrt {x} + \frac {2}{5} \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{2} + 2 \, A a b\right )} 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+a)^2/x^(1/2),x, algorithm="maxima")

[Out]

2/11*B*c^2*x^(11/2) + 2/9*(2*B*b*c + A*c^2)*x^(9/2) + 2/7*(B*b^2 + 2*(B*a + A*b)*c)*x^(7/2) + 2*A*a^2*sqrt(x)
+ 2/5*(2*B*a*b + A*b^2 + 2*A*a*c)*x^(5/2) + 2/3*(B*a^2 + 2*A*a*b)*x^(3/2)

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mupad [B]  time = 0.03, size = 93, normalized size = 0.84 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^2}{3}+\frac {4\,A\,b\,a}{3}\right )+x^{9/2}\,\left (\frac {2\,A\,c^2}{9}+\frac {4\,B\,b\,c}{9}\right )+x^{5/2}\,\left (\frac {2\,A\,b^2}{5}+\frac {4\,B\,a\,b}{5}+\frac {4\,A\,a\,c}{5}\right )+x^{7/2}\,\left (\frac {2\,B\,b^2}{7}+\frac {4\,A\,c\,b}{7}+\frac {4\,B\,a\,c}{7}\right )+2\,A\,a^2\,\sqrt {x}+\frac {2\,B\,c^2\,x^{11/2}}{11} \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^(1/2),x)

[Out]

x^(3/2)*((2*B*a^2)/3 + (4*A*a*b)/3) + x^(9/2)*((2*A*c^2)/9 + (4*B*b*c)/9) + x^(5/2)*((2*A*b^2)/5 + (4*A*a*c)/5
+ (4*B*a*b)/5) + x^(7/2)*((2*B*b^2)/7 + (4*A*b*c)/7 + (4*B*a*c)/7) + 2*A*a^2*x^(1/2) + (2*B*c^2*x^(11/2))/11

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sympy [A]  time = 2.05, size = 160, normalized size = 1.44 \begin {gather*} 2 A a^{2} \sqrt {x} + \frac {4 A a b x^{\frac {3}{2}}}{3} + \frac {4 A a c x^{\frac {5}{2}}}{5} + \frac {2 A b^{2} x^{\frac {5}{2}}}{5} + \frac {4 A b c x^{\frac {7}{2}}}{7} + \frac {2 A c^{2} x^{\frac {9}{2}}}{9} + \frac {2 B a^{2} x^{\frac {3}{2}}}{3} + \frac {4 B a b x^{\frac {5}{2}}}{5} + \frac {4 B a c x^{\frac {7}{2}}}{7} + \frac {2 B b^{2} x^{\frac {7}{2}}}{7} + \frac {4 B b c x^{\frac {9}{2}}}{9} + \frac {2 B c^{2} x^{\frac {11}{2}}}{11} \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**(1/2),x)

[Out]

2*A*a**2*sqrt(x) + 4*A*a*b*x**(3/2)/3 + 4*A*a*c*x**(5/2)/5 + 2*A*b**2*x**(5/2)/5 + 4*A*b*c*x**(7/2)/7 + 2*A*c*
*2*x**(9/2)/9 + 2*B*a**2*x**(3/2)/3 + 4*B*a*b*x**(5/2)/5 + 4*B*a*c*x**(7/2)/7 + 2*B*b**2*x**(7/2)/7 + 4*B*b*c*
x**(9/2)/9 + 2*B*c**2*x**(11/2)/11

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