3.8.31 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{\sqrt {x}} \, dx\)

Optimal. Leaf size=316 \[ \frac {20 a^2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 b^4 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {10 a b^3 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 a^4 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac {2 a^3 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

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Rubi [A]  time = 0.12, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} \frac {2 b^4 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {10 a b^3 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac {20 a^2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a^3 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x}+\frac {2 a^4 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[x],x]

[Out]

(2*a^5*A*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*a^4*(5*A*b + a*B)*x^(3/2)*Sqrt[a^2 + 2*a*b*x +
b^2*x^2])/(3*(a + b*x)) + (2*a^3*b*(2*A*b + a*B)*x^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^2*b^
2*(A*b + a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x)) + (10*a*b^3*(A*b + 2*a*B)*x^(9/2)*Sqrt[a^2
+ 2*a*b*x + b^2*x^2])/(9*(a + b*x)) + (2*b^4*(A*b + 5*a*B)*x^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*(a + b*
x)) + (2*b^5*B*x^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*(a + b*x))

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])

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{\sqrt {x}} \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^5 A b^5}{\sqrt {x}}+a^4 b^5 (5 A b+a B) \sqrt {x}+5 a^3 b^6 (2 A b+a B) x^{3/2}+10 a^2 b^7 (A b+a B) x^{5/2}+5 a b^8 (A b+2 a B) x^{7/2}+b^9 (A b+5 a B) x^{9/2}+b^{10} B x^{11/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 a^4 (5 A b+a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^3 b (2 A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {20 a^2 b^2 (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {10 a b^3 (A b+2 a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 126, normalized size = 0.40 \begin {gather*} \frac {2 \sqrt {x} \sqrt {(a+b x)^2} \left (3003 a^5 (3 A+B x)+3003 a^4 b x (5 A+3 B x)+2574 a^3 b^2 x^2 (7 A+5 B x)+1430 a^2 b^3 x^3 (9 A+7 B x)+455 a b^4 x^4 (11 A+9 B x)+63 b^5 x^5 (13 A+11 B x)\right )}{9009 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[x],x]

[Out]

(2*Sqrt[x]*Sqrt[(a + b*x)^2]*(3003*a^5*(3*A + B*x) + 3003*a^4*b*x*(5*A + 3*B*x) + 2574*a^3*b^2*x^2*(7*A + 5*B*
x) + 1430*a^2*b^3*x^3*(9*A + 7*B*x) + 455*a*b^4*x^4*(11*A + 9*B*x) + 63*b^5*x^5*(13*A + 11*B*x)))/(9009*(a + b
*x))

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IntegrateAlgebraic [A]  time = 10.45, size = 171, normalized size = 0.54 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (9009 a^5 A \sqrt {x}+3003 a^5 B x^{3/2}+15015 a^4 A b x^{3/2}+9009 a^4 b B x^{5/2}+18018 a^3 A b^2 x^{5/2}+12870 a^3 b^2 B x^{7/2}+12870 a^2 A b^3 x^{7/2}+10010 a^2 b^3 B x^{9/2}+5005 a A b^4 x^{9/2}+4095 a b^4 B x^{11/2}+819 A b^5 x^{11/2}+693 b^5 B x^{13/2}\right )}{9009 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/Sqrt[x],x]

[Out]

(2*Sqrt[(a + b*x)^2]*(9009*a^5*A*Sqrt[x] + 15015*a^4*A*b*x^(3/2) + 3003*a^5*B*x^(3/2) + 18018*a^3*A*b^2*x^(5/2
) + 9009*a^4*b*B*x^(5/2) + 12870*a^2*A*b^3*x^(7/2) + 12870*a^3*b^2*B*x^(7/2) + 5005*a*A*b^4*x^(9/2) + 10010*a^
2*b^3*B*x^(9/2) + 819*A*b^5*x^(11/2) + 4095*a*b^4*B*x^(11/2) + 693*b^5*B*x^(13/2)))/(9009*(a + b*x))

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fricas [A]  time = 0.42, size = 119, normalized size = 0.38 \begin {gather*} \frac {2}{9009} \, {\left (693 \, B b^{5} x^{6} + 9009 \, A a^{5} + 819 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 5005 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 12870 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 9009 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} 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)^(5/2)/x^(1/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^5*x^6 + 9009*A*a^5 + 819*(5*B*a*b^4 + A*b^5)*x^5 + 5005*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 12870*(B
*a^3*b^2 + A*a^2*b^3)*x^3 + 9009*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 3003*(B*a^5 + 5*A*a^4*b)*x)*sqrt(x)

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giac [A]  time = 0.17, size = 197, normalized size = 0.62 \begin {gather*} \frac {2}{13} \, B b^{5} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, B a b^{4} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, A b^{5} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, B a^{2} b^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a b^{4} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, B a^{3} b^{2} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, A a^{2} b^{3} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{4} b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{3} b^{2} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{4} b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) \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)^(5/2)/x^(1/2),x, algorithm="giac")

[Out]

2/13*B*b^5*x^(13/2)*sgn(b*x + a) + 10/11*B*a*b^4*x^(11/2)*sgn(b*x + a) + 2/11*A*b^5*x^(11/2)*sgn(b*x + a) + 20
/9*B*a^2*b^3*x^(9/2)*sgn(b*x + a) + 10/9*A*a*b^4*x^(9/2)*sgn(b*x + a) + 20/7*B*a^3*b^2*x^(7/2)*sgn(b*x + a) +
20/7*A*a^2*b^3*x^(7/2)*sgn(b*x + a) + 2*B*a^4*b*x^(5/2)*sgn(b*x + a) + 4*A*a^3*b^2*x^(5/2)*sgn(b*x + a) + 2/3*
B*a^5*x^(3/2)*sgn(b*x + a) + 10/3*A*a^4*b*x^(3/2)*sgn(b*x + a) + 2*A*a^5*sqrt(x)*sgn(b*x + a)

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maple [A]  time = 0.05, size = 140, normalized size = 0.44 \begin {gather*} \frac {2 \left (693 B \,b^{5} x^{6}+819 A \,b^{5} x^{5}+4095 B a \,b^{4} x^{5}+5005 A a \,b^{4} x^{4}+10010 B \,a^{2} b^{3} x^{4}+12870 A \,a^{2} b^{3} x^{3}+12870 B \,a^{3} b^{2} x^{3}+18018 A \,a^{3} b^{2} x^{2}+9009 B \,a^{4} b \,x^{2}+15015 A \,a^{4} b x +3003 B \,a^{5} x +9009 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \sqrt {x}}{9009 \left (b x +a \right )^{5}} \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)^(5/2)/x^(1/2),x)

[Out]

2/9009*x^(1/2)*(693*B*b^5*x^6+819*A*b^5*x^5+4095*B*a*b^4*x^5+5005*A*a*b^4*x^4+10010*B*a^2*b^3*x^4+12870*A*a^2*
b^3*x^3+12870*B*a^3*b^2*x^3+18018*A*a^3*b^2*x^2+9009*B*a^4*b*x^2+15015*A*a^4*b*x+3003*B*a^5*x+9009*A*a^5)*((b*
x+a)^2)^(5/2)/(b*x+a)^5

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maxima [A]  time = 0.77, size = 240, normalized size = 0.76 \begin {gather*} \frac {2}{3465} \, {\left (35 \, {\left (9 \, b^{5} x^{2} + 11 \, a b^{4} x\right )} x^{\frac {7}{2}} + 220 \, {\left (7 \, a b^{4} x^{2} + 9 \, a^{2} b^{3} x\right )} x^{\frac {5}{2}} + 594 \, {\left (5 \, a^{2} b^{3} x^{2} + 7 \, a^{3} b^{2} x\right )} x^{\frac {3}{2}} + 924 \, {\left (3 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x\right )} \sqrt {x} + \frac {1155 \, {\left (a^{4} b x^{2} + 3 \, a^{5} x\right )}}{\sqrt {x}}\right )} A + \frac {2}{45045} \, {\left (315 \, {\left (11 \, b^{5} x^{2} + 13 \, a b^{4} x\right )} x^{\frac {9}{2}} + 1820 \, {\left (9 \, a b^{4} x^{2} + 11 \, a^{2} b^{3} x\right )} x^{\frac {7}{2}} + 4290 \, {\left (7 \, a^{2} b^{3} x^{2} + 9 \, a^{3} b^{2} x\right )} x^{\frac {5}{2}} + 5148 \, {\left (5 \, a^{3} b^{2} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {3}{2}} + 3003 \, {\left (3 \, a^{4} b x^{2} + 5 \, a^{5} x\right )} \sqrt {x}\right )} B \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)^(5/2)/x^(1/2),x, algorithm="maxima")

[Out]

2/3465*(35*(9*b^5*x^2 + 11*a*b^4*x)*x^(7/2) + 220*(7*a*b^4*x^2 + 9*a^2*b^3*x)*x^(5/2) + 594*(5*a^2*b^3*x^2 + 7
*a^3*b^2*x)*x^(3/2) + 924*(3*a^3*b^2*x^2 + 5*a^4*b*x)*sqrt(x) + 1155*(a^4*b*x^2 + 3*a^5*x)/sqrt(x))*A + 2/4504
5*(315*(11*b^5*x^2 + 13*a*b^4*x)*x^(9/2) + 1820*(9*a*b^4*x^2 + 11*a^2*b^3*x)*x^(7/2) + 4290*(7*a^2*b^3*x^2 + 9
*a^3*b^2*x)*x^(5/2) + 5148*(5*a^3*b^2*x^2 + 7*a^4*b*x)*x^(3/2) + 3003*(3*a^4*b*x^2 + 5*a^5*x)*sqrt(x))*B

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{\sqrt {x}} \,d x \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)^(5/2))/x^(1/2),x)

[Out]

int(((A + B*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/x^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\sqrt {x}}\, dx \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)**(5/2)/x**(1/2),x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/sqrt(x), x)

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