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

Optimal. Leaf size=109 \[ 2 a^4 A \sqrt {x}+\frac {2}{3} a^3 x^{3/2} (a B+4 A b)+\frac {4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac {2}{9} b^3 x^{9/2} (4 a B+A b)+\frac {4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac {2}{11} b^4 B x^{11/2} \]

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Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 76} \begin {gather*} \frac {4}{5} a^2 b x^{5/2} (2 a B+3 A b)+\frac {2}{3} a^3 x^{3/2} (a B+4 A b)+2 a^4 A \sqrt {x}+\frac {2}{9} b^3 x^{9/2} (4 a B+A b)+\frac {4}{7} a b^2 x^{7/2} (3 a B+2 A b)+\frac {2}{11} b^4 B x^{11/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*a^4*A*Sqrt[x] + (2*a^3*(4*A*b + a*B)*x^(3/2))/3 + (4*a^2*b*(3*A*b + 2*a*B)*x^(5/2))/5 + (4*a*b^2*(2*A*b + 3*
a*B)*x^(7/2))/7 + (2*b^3*(A*b + 4*a*B)*x^(9/2))/9 + (2*b^4*B*x^(11/2))/11

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {x}} \, dx &=\int \frac {(a+b x)^4 (A+B x)}{\sqrt {x}} \, dx\\ &=\int \left (\frac {a^4 A}{\sqrt {x}}+a^3 (4 A b+a B) \sqrt {x}+2 a^2 b (3 A b+2 a B) x^{3/2}+2 a b^2 (2 A b+3 a B) x^{5/2}+b^3 (A b+4 a B) x^{7/2}+b^4 B x^{9/2}\right ) \, dx\\ &=2 a^4 A \sqrt {x}+\frac {2}{3} a^3 (4 A b+a B) x^{3/2}+\frac {4}{5} a^2 b (3 A b+2 a B) x^{5/2}+\frac {4}{7} a b^2 (2 A b+3 a B) x^{7/2}+\frac {2}{9} b^3 (A b+4 a B) x^{9/2}+\frac {2}{11} b^4 B x^{11/2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 89, normalized size = 0.82 \begin {gather*} \frac {2 \sqrt {x} \left (1155 a^4 (3 A+B x)+924 a^3 b x (5 A+3 B x)+594 a^2 b^2 x^2 (7 A+5 B x)+220 a b^3 x^3 (9 A+7 B x)+35 b^4 x^4 (11 A+9 B x)\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(1155*a^4*(3*A + B*x) + 924*a^3*b*x*(5*A + 3*B*x) + 594*a^2*b^2*x^2*(7*A + 5*B*x) + 220*a*b^3*x^3*(
9*A + 7*B*x) + 35*b^4*x^4*(11*A + 9*B*x)))/3465

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IntegrateAlgebraic [A]  time = 0.05, size = 125, normalized size = 1.15 \begin {gather*} \frac {2 \left (3465 a^4 A \sqrt {x}+1155 a^4 B x^{3/2}+4620 a^3 A b x^{3/2}+2772 a^3 b B x^{5/2}+4158 a^2 A b^2 x^{5/2}+2970 a^2 b^2 B x^{7/2}+1980 a A b^3 x^{7/2}+1540 a b^3 B x^{9/2}+385 A b^4 x^{9/2}+315 b^4 B x^{11/2}\right )}{3465} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3465*a^4*A*Sqrt[x] + 4620*a^3*A*b*x^(3/2) + 1155*a^4*B*x^(3/2) + 4158*a^2*A*b^2*x^(5/2) + 2772*a^3*b*B*x^(
5/2) + 1980*a*A*b^3*x^(7/2) + 2970*a^2*b^2*B*x^(7/2) + 385*A*b^4*x^(9/2) + 1540*a*b^3*B*x^(9/2) + 315*b^4*B*x^
(11/2)))/3465

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fricas [A]  time = 0.43, size = 99, normalized size = 0.91 \begin {gather*} \frac {2}{3465} \, {\left (315 \, B b^{4} x^{5} + 3465 \, A a^{4} + 385 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 990 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 1386 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 1155 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^4*x^5 + 3465*A*a^4 + 385*(4*B*a*b^3 + A*b^4)*x^4 + 990*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 1386*(2
*B*a^3*b + 3*A*a^2*b^2)*x^2 + 1155*(B*a^4 + 4*A*a^3*b)*x)*sqrt(x)

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giac [A]  time = 0.15, size = 101, normalized size = 0.93 \begin {gather*} \frac {2}{11} \, B b^{4} x^{\frac {11}{2}} + \frac {8}{9} \, B a b^{3} x^{\frac {9}{2}} + \frac {2}{9} \, A b^{4} x^{\frac {9}{2}} + \frac {12}{7} \, B a^{2} b^{2} x^{\frac {7}{2}} + \frac {8}{7} \, A a b^{3} x^{\frac {7}{2}} + \frac {8}{5} \, B a^{3} b x^{\frac {5}{2}} + \frac {12}{5} \, A a^{2} b^{2} x^{\frac {5}{2}} + \frac {2}{3} \, B a^{4} x^{\frac {3}{2}} + \frac {8}{3} \, A a^{3} b x^{\frac {3}{2}} + 2 \, A a^{4} \sqrt {x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(1/2),x, algorithm="giac")

[Out]

2/11*B*b^4*x^(11/2) + 8/9*B*a*b^3*x^(9/2) + 2/9*A*b^4*x^(9/2) + 12/7*B*a^2*b^2*x^(7/2) + 8/7*A*a*b^3*x^(7/2) +
8/5*B*a^3*b*x^(5/2) + 12/5*A*a^2*b^2*x^(5/2) + 2/3*B*a^4*x^(3/2) + 8/3*A*a^3*b*x^(3/2) + 2*A*a^4*sqrt(x)

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maple [A]  time = 0.05, size = 100, normalized size = 0.92 \begin {gather*} \frac {2 \left (315 b^{4} B \,x^{5}+385 A \,b^{4} x^{4}+1540 x^{4} B a \,b^{3}+1980 A a \,b^{3} x^{3}+2970 B \,a^{2} b^{2} x^{3}+4158 A \,a^{2} b^{2} x^{2}+2772 B \,a^{3} b \,x^{2}+4620 A \,a^{3} b x +1155 B \,a^{4} x +3465 A \,a^{4}\right ) \sqrt {x}}{3465} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(1/2),x)

[Out]

2/3465*x^(1/2)*(315*B*b^4*x^5+385*A*b^4*x^4+1540*B*a*b^3*x^4+1980*A*a*b^3*x^3+2970*B*a^2*b^2*x^3+4158*A*a^2*b^
2*x^2+2772*B*a^3*b*x^2+4620*A*a^3*b*x+1155*B*a^4*x+3465*A*a^4)

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maxima [A]  time = 0.47, size = 99, normalized size = 0.91 \begin {gather*} \frac {2}{11} \, B b^{4} x^{\frac {11}{2}} + 2 \, A a^{4} \sqrt {x} + \frac {2}{9} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {9}{2}} + \frac {4}{7} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {7}{2}} + \frac {4}{5} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/x^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 91, normalized size = 0.83 \begin {gather*} x^{3/2}\,\left (\frac {2\,B\,a^4}{3}+\frac {8\,A\,b\,a^3}{3}\right )+x^{9/2}\,\left (\frac {2\,A\,b^4}{9}+\frac {8\,B\,a\,b^3}{9}\right )+2\,A\,a^4\,\sqrt {x}+\frac {2\,B\,b^4\,x^{11/2}}{11}+\frac {4\,a^2\,b\,x^{5/2}\,\left (3\,A\,b+2\,B\,a\right )}{5}+\frac {4\,a\,b^2\,x^{7/2}\,\left (2\,A\,b+3\,B\,a\right )}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^2)/x^(1/2),x)

[Out]

x^(3/2)*((2*B*a^4)/3 + (8*A*a^3*b)/3) + x^(9/2)*((2*A*b^4)/9 + (8*B*a*b^3)/9) + 2*A*a^4*x^(1/2) + (2*B*b^4*x^(
11/2))/11 + (4*a^2*b*x^(5/2)*(3*A*b + 2*B*a))/5 + (4*a*b^2*x^(7/2)*(2*A*b + 3*B*a))/7

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sympy [A]  time = 1.58, size = 146, normalized size = 1.34 \begin {gather*} 2 A a^{4} \sqrt {x} + \frac {8 A a^{3} b x^{\frac {3}{2}}}{3} + \frac {12 A a^{2} b^{2} x^{\frac {5}{2}}}{5} + \frac {8 A a b^{3} x^{\frac {7}{2}}}{7} + \frac {2 A b^{4} x^{\frac {9}{2}}}{9} + \frac {2 B a^{4} x^{\frac {3}{2}}}{3} + \frac {8 B a^{3} b x^{\frac {5}{2}}}{5} + \frac {12 B a^{2} b^{2} x^{\frac {7}{2}}}{7} + \frac {8 B a b^{3} x^{\frac {9}{2}}}{9} + \frac {2 B b^{4} x^{\frac {11}{2}}}{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/x**(1/2),x)

[Out]

2*A*a**4*sqrt(x) + 8*A*a**3*b*x**(3/2)/3 + 12*A*a**2*b**2*x**(5/2)/5 + 8*A*a*b**3*x**(7/2)/7 + 2*A*b**4*x**(9/
2)/9 + 2*B*a**4*x**(3/2)/3 + 8*B*a**3*b*x**(5/2)/5 + 12*B*a**2*b**2*x**(7/2)/7 + 8*B*a*b**3*x**(9/2)/9 + 2*B*b
**4*x**(11/2)/11

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