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

Optimal. Leaf size=296 \[ -\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}+\frac {5 a b^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (a+b x)}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^3 (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^2 (a+b x)} \]

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Rubi [A]  time = 0.13, antiderivative size = 296, 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 = {770, 76} \begin {gather*} -\frac {a^4 \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 x^3 (a+b x)}-\frac {5 a^3 b \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{2 x^2 (a+b x)}-\frac {10 a^2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{x (a+b x)}+\frac {b^4 x \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{a+b x}+\frac {5 a b^3 \log (x) \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{a+b x}-\frac {a^5 A \sqrt {a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}+\frac {b^5 B x^2 \sqrt {a^2+2 a b x+b^2 x^2}}{2 (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^5,x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 126, normalized size = 0.43 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^5 (3 A+4 B x)+10 a^4 b x (2 A+3 B x)+60 a^3 b^2 x^2 (A+2 B x)+120 a^2 A b^3 x^3-60 a b^3 x^4 \log (x) (2 a B+A b)-60 a b^4 B x^5-6 b^5 x^5 (2 A+B x)\right )}{12 x^4 (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^5,x]

[Out]

-1/12*(Sqrt[(a + b*x)^2]*(120*a^2*A*b^3*x^3 - 60*a*b^4*B*x^5 - 6*b^5*x^5*(2*A + B*x) + 60*a^3*b^2*x^2*(A + 2*B
*x) + 10*a^4*b*x*(2*A + 3*B*x) + a^5*(3*A + 4*B*x) - 60*a*b^3*(A*b + 2*a*B)*x^4*Log[x]))/(x^4*(a + b*x))

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IntegrateAlgebraic [B]  time = 2.90, size = 859, normalized size = 2.90 \begin {gather*} 5 a A \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) b^4+10 a^2 B \tanh ^{-1}\left (\frac {\sqrt {b^2} x}{a}-\frac {\sqrt {a^2+2 b x a+b^2 x^2}}{a}\right ) b^3-\frac {5}{2} a A \sqrt {b^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b^3-\frac {5}{2} a A \sqrt {b^2} \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) b^3+\frac {-\sqrt {a^2+2 b x a+b^2 x^2} \left (-12 B x^9 b^9-24 A x^8 b^9-156 a B x^8 b^8-84 a A x^7 b^8-453 a^2 B x^7 b^7+132 a^2 A x^6 b^7-303 a^3 B x^6 b^6+780 a^3 A x^5 b^6+489 a^4 B x^5 b^5+1108 a^4 A x^4 b^5+851 a^5 B x^4 b^4+726 a^5 A x^3 b^4+444 a^6 B x^3 b^3+258 a^6 A x^2 b^3+84 a^7 B x^2 b^2+58 a^7 A x b^2+6 a^8 A b+8 a^8 B x b\right ) b^3-\sqrt {b^2} \left (12 b^9 B x^{10}+24 A b^9 x^9+168 a b^8 B x^9+108 a A b^8 x^8+609 a^2 b^7 B x^8-48 a^2 A b^7 x^7+756 a^3 b^6 B x^7-912 a^3 A b^6 x^6-186 a^4 b^5 B x^6-1888 a^4 A b^5 x^5-1340 a^5 b^4 B x^5-1834 a^5 A b^4 x^4-1295 a^6 b^3 B x^4-984 a^6 A b^3 x^3-528 a^7 b^2 B x^3-316 a^7 A b^2 x^2-92 a^8 b B x^2-64 a^8 A b x-8 a^9 B x-6 a^9 A\right ) b^3}{3 \sqrt {b^2} \sqrt {a^2+2 b x a+b^2 x^2} \left (-8 x^3 b^6-24 a x^2 b^5-24 a^2 x b^4-8 a^3 b^3\right ) x^4+3 \left (8 x^4 b^8+32 a x^3 b^7+48 a^2 x^2 b^6+32 a^3 x b^5+8 a^4 b^4\right ) x^4}-5 a^2 \left (b^2\right )^{3/2} B \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right )-5 a^2 \left (b^2\right )^{3/2} B \log \left (a-\sqrt {b^2} x+\sqrt {a^2+2 b x a+b^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2]*(6*a^8*A*b + 58*a^7*A*b^2*x + 8*a^8*b*B*x + 258*a^6*A*b^3*x^2 + 84*a^7*b^
2*B*x^2 + 726*a^5*A*b^4*x^3 + 444*a^6*b^3*B*x^3 + 1108*a^4*A*b^5*x^4 + 851*a^5*b^4*B*x^4 + 780*a^3*A*b^6*x^5 +
489*a^4*b^5*B*x^5 + 132*a^2*A*b^7*x^6 - 303*a^3*b^6*B*x^6 - 84*a*A*b^8*x^7 - 453*a^2*b^7*B*x^7 - 24*A*b^9*x^8
- 156*a*b^8*B*x^8 - 12*b^9*B*x^9)) - b^3*Sqrt[b^2]*(-6*a^9*A - 64*a^8*A*b*x - 8*a^9*B*x - 316*a^7*A*b^2*x^2 -
92*a^8*b*B*x^2 - 984*a^6*A*b^3*x^3 - 528*a^7*b^2*B*x^3 - 1834*a^5*A*b^4*x^4 - 1295*a^6*b^3*B*x^4 - 1888*a^4*A
*b^5*x^5 - 1340*a^5*b^4*B*x^5 - 912*a^3*A*b^6*x^6 - 186*a^4*b^5*B*x^6 - 48*a^2*A*b^7*x^7 + 756*a^3*b^6*B*x^7 +
108*a*A*b^8*x^8 + 609*a^2*b^7*B*x^8 + 24*A*b^9*x^9 + 168*a*b^8*B*x^9 + 12*b^9*B*x^10))/(3*Sqrt[b^2]*x^4*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]*(-8*a^3*b^3 - 24*a^2*b^4*x - 24*a*b^5*x^2 - 8*b^6*x^3) + 3*x^4*(8*a^4*b^4 + 32*a^3*b^
5*x + 48*a^2*b^6*x^2 + 32*a*b^7*x^3 + 8*b^8*x^4)) + 5*a*A*b^4*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b
^2*x^2]/a] + 10*a^2*b^3*B*ArcTanh[(Sqrt[b^2]*x)/a - Sqrt[a^2 + 2*a*b*x + b^2*x^2]/a] - (5*a*A*b^3*Sqrt[b^2]*Lo
g[-a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/2 - 5*a^2*(b^2)^(3/2)*B*Log[-a - Sqrt[b^2]*x + Sqrt[a^2 +
2*a*b*x + b^2*x^2]] - (5*a*A*b^3*Sqrt[b^2]*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]])/2 - 5*a^2*(b
^2)^(3/2)*B*Log[a - Sqrt[b^2]*x + Sqrt[a^2 + 2*a*b*x + b^2*x^2]]

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fricas [A]  time = 0.42, size = 121, normalized size = 0.41 \begin {gather*} \frac {6 \, B b^{5} x^{6} - 3 \, A a^{5} + 12 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 60 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} \log \relax (x) - 120 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} - 30 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} - 4 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{12 \, x^{4}} \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^5,x, algorithm="fricas")

[Out]

1/12*(6*B*b^5*x^6 - 3*A*a^5 + 12*(5*B*a*b^4 + A*b^5)*x^5 + 60*(2*B*a^2*b^3 + A*a*b^4)*x^4*log(x) - 120*(B*a^3*
b^2 + A*a^2*b^3)*x^3 - 30*(B*a^4*b + 2*A*a^3*b^2)*x^2 - 4*(B*a^5 + 5*A*a^4*b)*x)/x^4

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giac [A]  time = 0.17, size = 188, normalized size = 0.64 \begin {gather*} \frac {1}{2} \, B b^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 5 \, B a b^{4} x \mathrm {sgn}\left (b x + a\right ) + A b^{5} x \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (2 \, B a^{2} b^{3} \mathrm {sgn}\left (b x + a\right ) + A a b^{4} \mathrm {sgn}\left (b x + a\right )\right )} \log \left ({\left | x \right |}\right ) - \frac {3 \, A a^{5} \mathrm {sgn}\left (b x + a\right ) + 120 \, {\left (B a^{3} b^{2} \mathrm {sgn}\left (b x + a\right ) + A a^{2} b^{3} \mathrm {sgn}\left (b x + a\right )\right )} x^{3} + 30 \, {\left (B a^{4} b \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{3} b^{2} \mathrm {sgn}\left (b x + a\right )\right )} x^{2} + 4 \, {\left (B a^{5} \mathrm {sgn}\left (b x + a\right ) + 5 \, A a^{4} b \mathrm {sgn}\left (b x + a\right )\right )} x}{12 \, x^{4}} \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^5,x, algorithm="giac")

[Out]

1/2*B*b^5*x^2*sgn(b*x + a) + 5*B*a*b^4*x*sgn(b*x + a) + A*b^5*x*sgn(b*x + a) + 5*(2*B*a^2*b^3*sgn(b*x + a) + A
*a*b^4*sgn(b*x + a))*log(abs(x)) - 1/12*(3*A*a^5*sgn(b*x + a) + 120*(B*a^3*b^2*sgn(b*x + a) + A*a^2*b^3*sgn(b*
x + a))*x^3 + 30*(B*a^4*b*sgn(b*x + a) + 2*A*a^3*b^2*sgn(b*x + a))*x^2 + 4*(B*a^5*sgn(b*x + a) + 5*A*a^4*b*sgn
(b*x + a))*x)/x^4

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maple [A]  time = 0.36, size = 144, normalized size = 0.49 \begin {gather*} \frac {\left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \left (6 B \,b^{5} x^{6}+60 A a \,b^{4} x^{4} \ln \relax (x )+12 A \,b^{5} x^{5}+120 B \,a^{2} b^{3} x^{4} \ln \relax (x )+60 B a \,b^{4} x^{5}-120 A \,a^{2} b^{3} x^{3}-120 B \,a^{3} b^{2} x^{3}-60 A \,a^{3} b^{2} x^{2}-30 B \,a^{4} b \,x^{2}-20 A \,a^{4} b x -4 B \,a^{5} x -3 A \,a^{5}\right )}{12 \left (b x +a \right )^{5} x^{4}} \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^5,x)

[Out]

1/12*((b*x+a)^2)^(5/2)*(6*B*b^5*x^6+60*A*ln(x)*x^4*a*b^4+12*A*b^5*x^5+120*B*ln(x)*x^4*a^2*b^3+60*B*a*b^4*x^5-1
20*A*a^2*b^3*x^3-120*B*a^3*b^2*x^3-60*A*a^3*b^2*x^2-30*B*a^4*b*x^2-20*A*a^4*b*x-4*B*a^5*x-3*A*a^5)/(b*x+a)^5/x
^4

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maxima [B]  time = 0.62, size = 615, normalized size = 2.08 \begin {gather*} 10 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} B a^{2} b^{3} \log \left (2 \, b^{2} x + 2 \, a b\right ) + 5 \, \left (-1\right )^{2 \, b^{2} x + 2 \, a b} A a b^{4} \log \left (2 \, b^{2} x + 2 \, a b\right ) - 10 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} B a^{2} b^{3} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) - 5 \, \left (-1\right )^{2 \, a b x + 2 \, a^{2}} A a b^{4} \log \left (\frac {2 \, a b x}{{\left | x \right |}} + \frac {2 \, a^{2}}{{\left | x \right |}}\right ) + 5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B b^{4} x + \frac {5 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{5} x}{2 \, a} + 15 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} B a b^{3} + \frac {15}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} A b^{4} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{4} x}{2 \, a^{2}} + \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{5} x}{4 \, a^{3}} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} B b^{3}}{6 \, a} + \frac {35 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} A b^{4}}{12 \, a^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{3}}{6 \, a^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{4}}{3 \, a^{4}} - \frac {11 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} B b^{2}}{6 \, a^{2} x} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}} A b^{3}}{3 \, a^{3} x} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B b}{6 \, a^{3} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b^{2}}{3 \, a^{4} x^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} B}{3 \, a^{2} x^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A b}{12 \, a^{3} x^{3}} - \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {7}{2}} A}{4 \, a^{2} x^{4}} \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^5,x, algorithm="maxima")

[Out]

10*(-1)^(2*b^2*x + 2*a*b)*B*a^2*b^3*log(2*b^2*x + 2*a*b) + 5*(-1)^(2*b^2*x + 2*a*b)*A*a*b^4*log(2*b^2*x + 2*a*
b) - 10*(-1)^(2*a*b*x + 2*a^2)*B*a^2*b^3*log(2*a*b*x/abs(x) + 2*a^2/abs(x)) - 5*(-1)^(2*a*b*x + 2*a^2)*A*a*b^4
*log(2*a*b*x/abs(x) + 2*a^2/abs(x)) + 5*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*b^4*x + 5/2*sqrt(b^2*x^2 + 2*a*b*x + a
^2)*A*b^5*x/a + 15*sqrt(b^2*x^2 + 2*a*b*x + a^2)*B*a*b^3 + 15/2*sqrt(b^2*x^2 + 2*a*b*x + a^2)*A*b^4 + 5/2*(b^2
*x^2 + 2*a*b*x + a^2)^(3/2)*B*b^4*x/a^2 + 5/4*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^5*x/a^3 + 35/6*(b^2*x^2 + 2*
a*b*x + a^2)^(3/2)*B*b^3/a + 35/12*(b^2*x^2 + 2*a*b*x + a^2)^(3/2)*A*b^4/a^2 + 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(
5/2)*B*b^3/a^3 + 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^4/a^4 - 11/6*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*B*b^2/(a
^2*x) - 2/3*(b^2*x^2 + 2*a*b*x + a^2)^(5/2)*A*b^3/(a^3*x) - 1/6*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B*b/(a^3*x^2)
- 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b^2/(a^4*x^2) - 1/3*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*B/(a^2*x^3) + 1/12
*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A*b/(a^3*x^3) - 1/4*(b^2*x^2 + 2*a*b*x + a^2)^(7/2)*A/(a^2*x^4)

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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}}{x^5} \,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^5,x)

[Out]

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

[Out]

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

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