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

Optimal. Leaf size=107 \[ -\frac {2 a^4 A}{\sqrt {x}}+2 a^3 \sqrt {x} (a B+4 A b)+\frac {4}{3} a^2 b x^{3/2} (2 a B+3 A b)+\frac {2}{7} b^3 x^{7/2} (4 a B+A b)+\frac {4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac {2}{9} b^4 B x^{9/2} \]

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Rubi [A]  time = 0.05, antiderivative size = 107, 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}{3} a^2 b x^{3/2} (2 a B+3 A b)+2 a^3 \sqrt {x} (a B+4 A b)-\frac {2 a^4 A}{\sqrt {x}}+\frac {2}{7} b^3 x^{7/2} (4 a B+A b)+\frac {4}{5} a b^2 x^{5/2} (3 a B+2 A b)+\frac {2}{9} b^4 B x^{9/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 87, normalized size = 0.81 \begin {gather*} \frac {-630 a^4 (A-B x)+840 a^3 b x (3 A+B x)+252 a^2 b^2 x^2 (5 A+3 B x)+72 a b^3 x^3 (7 A+5 B x)+10 b^4 x^4 (9 A+7 B x)}{315 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-630*a^4*(A - B*x) + 840*a^3*b*x*(3*A + B*x) + 252*a^2*b^2*x^2*(5*A + 3*B*x) + 72*a*b^3*x^3*(7*A + 5*B*x) + 1
0*b^4*x^4*(9*A + 7*B*x))/(315*Sqrt[x])

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IntegrateAlgebraic [A]  time = 0.07, size = 103, normalized size = 0.96 \begin {gather*} \frac {2 \left (-315 a^4 A+315 a^4 B x+1260 a^3 A b x+420 a^3 b B x^2+630 a^2 A b^2 x^2+378 a^2 b^2 B x^3+252 a A b^3 x^3+180 a b^3 B x^4+45 A b^4 x^4+35 b^4 B x^5\right )}{315 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-315*a^4*A + 1260*a^3*A*b*x + 315*a^4*B*x + 630*a^2*A*b^2*x^2 + 420*a^3*b*B*x^2 + 252*a*A*b^3*x^3 + 378*a^
2*b^2*B*x^3 + 45*A*b^4*x^4 + 180*a*b^3*B*x^4 + 35*b^4*B*x^5))/(315*Sqrt[x])

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fricas [A]  time = 0.41, size = 99, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (35 \, B b^{4} x^{5} - 315 \, A a^{4} + 45 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{4} + 126 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3} + 210 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{2} + 315 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x\right )}}{315 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/315*(35*B*b^4*x^5 - 315*A*a^4 + 45*(4*B*a*b^3 + A*b^4)*x^4 + 126*(3*B*a^2*b^2 + 2*A*a*b^3)*x^3 + 210*(2*B*a^
3*b + 3*A*a^2*b^2)*x^2 + 315*(B*a^4 + 4*A*a^3*b)*x)/sqrt(x)

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giac [A]  time = 0.20, size = 101, normalized size = 0.94 \begin {gather*} \frac {2}{9} \, B b^{4} x^{\frac {9}{2}} + \frac {8}{7} \, B a b^{3} x^{\frac {7}{2}} + \frac {2}{7} \, A b^{4} x^{\frac {7}{2}} + \frac {12}{5} \, B a^{2} b^{2} x^{\frac {5}{2}} + \frac {8}{5} \, A a b^{3} x^{\frac {5}{2}} + \frac {8}{3} \, B a^{3} b x^{\frac {3}{2}} + 4 \, A a^{2} b^{2} x^{\frac {3}{2}} + 2 \, B a^{4} \sqrt {x} + 8 \, A a^{3} b \sqrt {x} - \frac {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^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 100, normalized size = 0.93 \begin {gather*} -\frac {2 \left (-35 b^{4} B \,x^{5}-45 A \,b^{4} x^{4}-180 x^{4} B a \,b^{3}-252 A a \,b^{3} x^{3}-378 B \,a^{2} b^{2} x^{3}-630 A \,a^{2} b^{2} x^{2}-420 B \,a^{3} b \,x^{2}-1260 A \,a^{3} b x -315 B \,a^{4} x +315 A \,a^{4}\right )}{315 \sqrt {x}} \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^(3/2),x)

[Out]

-2/315*(-35*B*b^4*x^5-45*A*b^4*x^4-180*B*a*b^3*x^4-252*A*a*b^3*x^3-378*B*a^2*b^2*x^3-630*A*a^2*b^2*x^2-420*B*a
^3*b*x^2-1260*A*a^3*b*x-315*B*a^4*x+315*A*a^4)/x^(1/2)

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maxima [A]  time = 0.62, size = 99, normalized size = 0.93 \begin {gather*} \frac {2}{9} \, B b^{4} x^{\frac {9}{2}} - \frac {2 \, A a^{4}}{\sqrt {x}} + \frac {2}{7} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {7}{2}} + \frac {4}{5} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {5}{2}} + \frac {4}{3} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {3}{2}} + 2 \, {\left (B a^{4} + 4 \, A a^{3} b\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^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.04, size = 91, normalized size = 0.85 \begin {gather*} \sqrt {x}\,\left (2\,B\,a^4+8\,A\,b\,a^3\right )+x^{7/2}\,\left (\frac {2\,A\,b^4}{7}+\frac {8\,B\,a\,b^3}{7}\right )-\frac {2\,A\,a^4}{\sqrt {x}}+\frac {2\,B\,b^4\,x^{9/2}}{9}+\frac {4\,a^2\,b\,x^{3/2}\,\left (3\,A\,b+2\,B\,a\right )}{3}+\frac {4\,a\,b^2\,x^{5/2}\,\left (2\,A\,b+3\,B\,a\right )}{5} \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^(3/2),x)

[Out]

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

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sympy [A]  time = 1.80, size = 141, normalized size = 1.32 \begin {gather*} - \frac {2 A a^{4}}{\sqrt {x}} + 8 A a^{3} b \sqrt {x} + 4 A a^{2} b^{2} x^{\frac {3}{2}} + \frac {8 A a b^{3} x^{\frac {5}{2}}}{5} + \frac {2 A b^{4} x^{\frac {7}{2}}}{7} + 2 B a^{4} \sqrt {x} + \frac {8 B a^{3} b x^{\frac {3}{2}}}{3} + \frac {12 B a^{2} b^{2} x^{\frac {5}{2}}}{5} + \frac {8 B a b^{3} x^{\frac {7}{2}}}{7} + \frac {2 B b^{4} x^{\frac {9}{2}}}{9} \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**(3/2),x)

[Out]

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

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