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

Optimal. Leaf size=314 \[ \frac {20 a^2 b^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac {2 b^4 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 (a+b x)}+\frac {2 a b^3 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac {2 b^5 B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{\sqrt {x} (a+b x)}+\frac {10 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]

[Out]

-2/3*a^5*A*((b*x+a)^2)^(1/2)/x^(3/2)/(b*x+a)+20/3*a^2*b^2*(A*b+B*a)*x^(3/2)*((b*x+a)^2)^(1/2)/(b*x+a)+2*a*b^3*
(A*b+2*B*a)*x^(5/2)*((b*x+a)^2)^(1/2)/(b*x+a)+2/7*b^4*(A*b+5*B*a)*x^(7/2)*((b*x+a)^2)^(1/2)/(b*x+a)+2/9*b^5*B*
x^(9/2)*((b*x+a)^2)^(1/2)/(b*x+a)-2*a^4*(5*A*b+B*a)*((b*x+a)^2)^(1/2)/(b*x+a)/x^(1/2)+10*a^3*b*(2*A*b+B*a)*x^(
1/2)*((b*x+a)^2)^(1/2)/(b*x+a)

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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} \[ -\frac {2 a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{\sqrt {x} (a+b x)}+\frac {10 a^3 b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x}+\frac {20 a^2 b^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{3 (a+b x)}+\frac {2 a b^3 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac {2 b^4 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{7 (a+b x)}-\frac {2 a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}+\frac {2 b^5 B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a^5*A*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^(3/2)*(a + b*x)) - (2*a^4*(5*A*b + a*B)*Sqrt[a^2 + 2*a*b*x + b^2
*x^2])/(Sqrt[x]*(a + b*x)) + (10*a^3*b*(2*A*b + a*B)*Sqrt[x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (20*a^
2*b^2*(A*b + a*B)*x^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*(a + b*x)) + (2*a*b^3*(A*b + 2*a*B)*x^(5/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])/(a + b*x) + (2*b^4*(A*b + 5*a*B)*x^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*(a + b*x))
+ (2*b^5*B*x^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(9*(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^{5/2}} \, 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^{5/2}} \, 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^{5/2}}+\frac {a^4 b^5 (5 A b+a B)}{x^{3/2}}+\frac {5 a^3 b^6 (2 A b+a B)}{\sqrt {x}}+10 a^2 b^7 (A b+a B) \sqrt {x}+5 a b^8 (A b+2 a B) x^{3/2}+b^9 (A b+5 a B) x^{5/2}+b^{10} B x^{7/2}\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}}{3 x^{3/2} (a+b x)}-\frac {2 a^4 (5 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {10 a^3 b (2 A b+a B) \sqrt {x} \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^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a b^3 (A b+2 a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^4 (A b+5 a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b^5 B x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 122, normalized size = 0.39 \[ -\frac {2 \sqrt {(a+b x)^2} \left (21 a^5 (A+3 B x)+315 a^4 b x (A-B x)-210 a^3 b^2 x^2 (3 A+B x)-42 a^2 b^3 x^3 (5 A+3 B x)-9 a b^4 x^4 (7 A+5 B x)-b^5 x^5 (9 A+7 B x)\right )}{63 x^{3/2} (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(a + b*x)^2]*(315*a^4*b*x*(A - B*x) - 210*a^3*b^2*x^2*(3*A + B*x) + 21*a^5*(A + 3*B*x) - 42*a^2*b^3*x
^3*(5*A + 3*B*x) - 9*a*b^4*x^4*(7*A + 5*B*x) - b^5*x^5*(9*A + 7*B*x)))/(63*x^(3/2)*(a + b*x))

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fricas [A]  time = 0.89, size = 119, normalized size = 0.38 \[ \frac {2 \, {\left (7 \, B b^{5} x^{6} - 21 \, A a^{5} + 9 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 63 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 210 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 315 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )}}{63 \, x^{\frac {3}{2}}} \]

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^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*B*b^5*x^6 - 21*A*a^5 + 9*(5*B*a*b^4 + A*b^5)*x^5 + 63*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 210*(B*a^3*b^2 + A
*a^2*b^3)*x^3 + 315*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 63*(B*a^5 + 5*A*a^4*b)*x)/x^(3/2)

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giac [A]  time = 0.18, size = 195, normalized size = 0.62 \[ \frac {2}{9} \, B b^{5} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a b^{4} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A b^{5} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 4 \, B a^{2} b^{3} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a b^{4} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, B a^{3} b^{2} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{3} \, A a^{2} b^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 10 \, B a^{4} b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 20 \, A a^{3} b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (3 \, B a^{5} x \mathrm {sgn}\left (b x + a\right ) + 15 \, A a^{4} b x \mathrm {sgn}\left (b x + a\right ) + A a^{5} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac {3}{2}}} \]

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^(5/2),x, algorithm="giac")

[Out]

2/9*B*b^5*x^(9/2)*sgn(b*x + a) + 10/7*B*a*b^4*x^(7/2)*sgn(b*x + a) + 2/7*A*b^5*x^(7/2)*sgn(b*x + a) + 4*B*a^2*
b^3*x^(5/2)*sgn(b*x + a) + 2*A*a*b^4*x^(5/2)*sgn(b*x + a) + 20/3*B*a^3*b^2*x^(3/2)*sgn(b*x + a) + 20/3*A*a^2*b
^3*x^(3/2)*sgn(b*x + a) + 10*B*a^4*b*sqrt(x)*sgn(b*x + a) + 20*A*a^3*b^2*sqrt(x)*sgn(b*x + a) - 2/3*(3*B*a^5*x
*sgn(b*x + a) + 15*A*a^4*b*x*sgn(b*x + a) + A*a^5*sgn(b*x + a))/x^(3/2)

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maple [A]  time = 0.06, size = 140, normalized size = 0.45 \[ -\frac {2 \left (-7 B \,b^{5} x^{6}-9 A \,b^{5} x^{5}-45 B a \,b^{4} x^{5}-63 A a \,b^{4} x^{4}-126 B \,a^{2} b^{3} x^{4}-210 A \,a^{2} b^{3} x^{3}-210 B \,a^{3} b^{2} x^{3}-630 A \,a^{3} b^{2} x^{2}-315 B \,a^{4} b \,x^{2}+315 A \,a^{4} b x +63 B \,a^{5} x +21 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{63 \left (b x +a \right )^{5} x^{\frac {3}{2}}} \]

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^(5/2),x)

[Out]

-2/63*(-7*B*b^5*x^6-9*A*b^5*x^5-45*B*a*b^4*x^5-63*A*a*b^4*x^4-126*B*a^2*b^3*x^4-210*A*a^2*b^3*x^3-210*B*a^3*b^
2*x^3-630*A*a^3*b^2*x^2-315*B*a^4*b*x^2+315*A*a^4*b*x+63*B*a^5*x+21*A*a^5)*((b*x+a)^2)^(5/2)/x^(3/2)/(b*x+a)^5

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maxima [A]  time = 0.57, size = 236, normalized size = 0.75 \[ \frac {2}{105} \, {\left (3 \, {\left (5 \, b^{5} x^{2} + 7 \, a b^{4} x\right )} x^{\frac {3}{2}} + 28 \, {\left (3 \, a b^{4} x^{2} + 5 \, a^{2} b^{3} x\right )} \sqrt {x} + \frac {210 \, {\left (a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x\right )}}{\sqrt {x}} + \frac {420 \, {\left (a^{3} b^{2} x^{2} - a^{4} b x\right )}}{x^{\frac {3}{2}}} - \frac {35 \, {\left (3 \, a^{4} b x^{2} + a^{5} x\right )}}{x^{\frac {5}{2}}}\right )} A + \frac {2}{315} \, {\left (5 \, {\left (7 \, b^{5} x^{2} + 9 \, a b^{4} x\right )} x^{\frac {5}{2}} + 36 \, {\left (5 \, a b^{4} x^{2} + 7 \, a^{2} b^{3} x\right )} x^{\frac {3}{2}} + 126 \, {\left (3 \, a^{2} b^{3} x^{2} + 5 \, a^{3} b^{2} x\right )} \sqrt {x} + \frac {420 \, {\left (a^{3} b^{2} x^{2} + 3 \, a^{4} b x\right )}}{\sqrt {x}} + \frac {315 \, {\left (a^{4} b x^{2} - a^{5} x\right )}}{x^{\frac {3}{2}}}\right )} B \]

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^(5/2),x, algorithm="maxima")

[Out]

2/105*(3*(5*b^5*x^2 + 7*a*b^4*x)*x^(3/2) + 28*(3*a*b^4*x^2 + 5*a^2*b^3*x)*sqrt(x) + 210*(a^2*b^3*x^2 + 3*a^3*b
^2*x)/sqrt(x) + 420*(a^3*b^2*x^2 - a^4*b*x)/x^(3/2) - 35*(3*a^4*b*x^2 + a^5*x)/x^(5/2))*A + 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))*B

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

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^(5/2),x)

[Out]

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

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

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**(5/2),x)

[Out]

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

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