3.8.32 \(\int \frac {(A+B x) (a^2+2 a b x+b^2 x^2)^{5/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=314 \[ \frac {4 a^2 b^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {2 b^4 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{9 (a+b x)}+\frac {10 a b^3 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 (a+b x)}+\frac {2 b^5 B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{a+b x}+\frac {10 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 (a+b x)} \]

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Rubi [A]  time = 0.12, antiderivative size = 314, 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^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{9 (a+b x)}+\frac {10 a b^3 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 (a+b x)}+\frac {4 a^2 b^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {10 a^3 b x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 (a+b x)}+\frac {2 a^4 \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{a+b x}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {2 b^5 B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (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))/x^(3/2),x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 124, normalized size = 0.39 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-693 a^5 (A-B x)+1155 a^4 b x (3 A+B x)+462 a^3 b^2 x^2 (5 A+3 B x)+198 a^2 b^3 x^3 (7 A+5 B x)+55 a b^4 x^4 (9 A+7 B x)+7 b^5 x^5 (11 A+9 B x)\right )}{693 \sqrt {x} (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))/x^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-693*a^5*(A - B*x) + 1155*a^4*b*x*(3*A + B*x) + 462*a^3*b^2*x^2*(5*A + 3*B*x) + 198*a^2*
b^3*x^3*(7*A + 5*B*x) + 55*a*b^4*x^4*(9*A + 7*B*x) + 7*b^5*x^5*(11*A + 9*B*x)))/(693*Sqrt[x]*(a + b*x))

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IntegrateAlgebraic [A]  time = 9.36, size = 145, normalized size = 0.46 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (-693 a^5 A+693 a^5 B x+3465 a^4 A b x+1155 a^4 b B x^2+2310 a^3 A b^2 x^2+1386 a^3 b^2 B x^3+1386 a^2 A b^3 x^3+990 a^2 b^3 B x^4+495 a A b^4 x^4+385 a b^4 B x^5+77 A b^5 x^5+63 b^5 B x^6\right )}{693 \sqrt {x} (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))/x^(3/2),x]

[Out]

(2*Sqrt[(a + b*x)^2]*(-693*a^5*A + 3465*a^4*A*b*x + 693*a^5*B*x + 2310*a^3*A*b^2*x^2 + 1155*a^4*b*B*x^2 + 1386
*a^2*A*b^3*x^3 + 1386*a^3*b^2*B*x^3 + 495*a*A*b^4*x^4 + 990*a^2*b^3*B*x^4 + 77*A*b^5*x^5 + 385*a*b^4*B*x^5 + 6
3*b^5*B*x^6))/(693*Sqrt[x]*(a + b*x))

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fricas [A]  time = 0.47, size = 119, normalized size = 0.38 \begin {gather*} \frac {2 \, {\left (63 \, B b^{5} x^{6} - 693 \, A a^{5} + 77 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 495 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 1386 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1155 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 693 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{693 \, \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^(3/2),x, algorithm="fricas")

[Out]

2/693*(63*B*b^5*x^6 - 693*A*a^5 + 77*(5*B*a*b^4 + A*b^5)*x^5 + 495*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 1386*(B*a^3*b
^2 + A*a^2*b^3)*x^3 + 1155*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 693*(B*a^5 + 5*A*a^4*b)*x)/sqrt(x)

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giac [A]  time = 0.18, size = 197, normalized size = 0.63 \begin {gather*} \frac {2}{11} \, B b^{5} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, B a b^{4} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A b^{5} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, B a^{2} b^{3} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, A a b^{4} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{3} b^{2} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{2} b^{3} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, B a^{4} b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, A a^{3} b^{2} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 10 \, A a^{4} b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, A a^{5} \mathrm {sgn}\left (b x + a\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^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 140, normalized size = 0.45 \begin {gather*} -\frac {2 \left (-63 B \,b^{5} x^{6}-77 A \,b^{5} x^{5}-385 B a \,b^{4} x^{5}-495 A a \,b^{4} x^{4}-990 B \,a^{2} b^{3} x^{4}-1386 A \,a^{2} b^{3} x^{3}-1386 B \,a^{3} b^{2} x^{3}-2310 A \,a^{3} b^{2} x^{2}-1155 B \,a^{4} b \,x^{2}-3465 A \,a^{4} b x -693 B \,a^{5} x +693 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{693 \left (b x +a \right )^{5} \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)^(5/2)/x^(3/2),x)

[Out]

-2/693*(-63*B*b^5*x^6-77*A*b^5*x^5-385*B*a*b^4*x^5-495*A*a*b^4*x^4-990*B*a^2*b^3*x^4-1386*A*a^2*b^3*x^3-1386*B
*a^3*b^2*x^3-2310*A*a^3*b^2*x^2-1155*B*a^4*b*x^2-3465*A*a^4*b*x-693*B*a^5*x+693*A*a^5)*((b*x+a)^2)^(5/2)/x^(1/
2)/(b*x+a)^5

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maxima [A]  time = 0.64, size = 238, normalized size = 0.76 \begin {gather*} \frac {2}{315} \, {\left (5 \, {\left (7 \, b^{5} x^{2} + 9 \, a b^{4} x\right )} x^{\frac {5}{2}} + 36 \, {\left (5 \, a b^{4} x^{2} + 7 \, a^{2} b^{3} x\right )} x^{\frac {3}{2}} + 126 \, {\left (3 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} x\right )} \sqrt {x} + \frac {420 \, {\left (a^{3} b^{2} x^{2} + 3 \, a^{4} b x\right )}}{\sqrt {x}} + \frac {315 \, {\left (a^{4} b x^{2} - a^{5} x\right )}}{x^{\frac {3}{2}}}\right )} A + \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 )} 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^(3/2),x, algorithm="maxima")

[Out]

2/315*(5*(7*b^5*x^2 + 9*a*b^4*x)*x^(5/2) + 36*(5*a*b^4*x^2 + 7*a^2*b^3*x)*x^(3/2) + 126*(3*a^2*b^3*x^2 + 5*a^3
*b^2*x)*sqrt(x) + 420*(a^3*b^2*x^2 + 3*a^4*b*x)/sqrt(x) + 315*(a^4*b*x^2 - a^5*x)/x^(3/2))*A + 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))*B

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mupad [B]  time = 1.85, size = 140, normalized size = 0.45 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {2\,B\,b^4\,x^6}{11}-\frac {2\,A\,a^5}{b}+\frac {10\,a^3\,x^2\,\left (2\,A\,b+B\,a\right )}{3}+\frac {x^5\,\left (154\,A\,b^5+770\,B\,a\,b^4\right )}{693\,b}+4\,a^2\,b\,x^3\,\left (A\,b+B\,a\right )+\frac {10\,a\,b^2\,x^4\,\left (A\,b+2\,B\,a\right )}{7}+\frac {2\,a^4\,x\,\left (5\,A\,b+B\,a\right )}{b}\right )}{x^{3/2}+\frac {a\,\sqrt {x}}{b}} \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^(3/2),x)

[Out]

((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((2*B*b^4*x^6)/11 - (2*A*a^5)/b + (10*a^3*x^2*(2*A*b + B*a))/3 + (x^5*(154*A*
b^5 + 770*B*a*b^4))/(693*b) + 4*a^2*b*x^3*(A*b + B*a) + (10*a*b^2*x^4*(A*b + 2*B*a))/7 + (2*a^4*x*(5*A*b + B*a
))/b))/(x^(3/2) + (a*x^(1/2))/b)

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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}}}{x^{\frac {3}{2}}}\, 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**(3/2),x)

[Out]

Integral((A + B*x)*((a + b*x)**2)**(5/2)/x**(3/2), x)

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