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

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

[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))

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Rubi [A]  time = 0.118602, antiderivative size = 314, normalized size of antiderivative = 1., 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} \[ \frac{2 b^4 x^{9/2} \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{9 (a+b x)}+\frac{10 a b^3 x^{7/2} \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{7 (a+b x)}+\frac{4 a^2 b^2 x^{5/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac{10 a^3 b x^{3/2} \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{3 (a+b x)}+\frac{2 a^4 \sqrt{x} \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{a+b x}-\frac{2 a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{\sqrt{x} (a+b x)}+\frac{2 b^5 B x^{11/2} \sqrt{a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]

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 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]

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 )^{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*}

Mathematica [A]  time = 0.0472925, size = 124, normalized size = 0.39 \[ \frac{2 \sqrt{(a+b x)^2} \left (462 a^3 b^2 x^2 (5 A+3 B x)+198 a^2 b^3 x^3 (7 A+5 B x)+1155 a^4 b x (3 A+B x)-693 a^5 (A-B x)+55 a b^4 x^4 (9 A+7 B x)+7 b^5 x^5 (11 A+9 B x)\right )}{693 \sqrt{x} (a+b x)} \]

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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Maple [A]  time = 0.006, size = 140, normalized size = 0.5 \begin{align*} -{\frac{-126\,B{b}^{5}{x}^{6}-154\,A{x}^{5}{b}^{5}-770\,B{x}^{5}a{b}^{4}-990\,A{x}^{4}a{b}^{4}-1980\,B{x}^{4}{a}^{2}{b}^{3}-2772\,A{x}^{3}{a}^{2}{b}^{3}-2772\,B{x}^{3}{a}^{3}{b}^{2}-4620\,A{x}^{2}{a}^{3}{b}^{2}-2310\,B{x}^{2}{a}^{4}b-6930\,A{a}^{4}bx-1386\,B{a}^{5}x+1386\,A{a}^{5}}{693\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}

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 = 1.09881, size = 321, normalized size = 1.02 \begin{align*} \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{align*}

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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Fricas [A]  time = 1.4104, size = 274, normalized size = 0.87 \begin{align*} \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{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [A]  time = 1.15544, size = 266, normalized size = 0.85 \begin{align*} \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{align*}

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)