3.421 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx\)

Optimal. Leaf size=208 \[ \frac {b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}}-\frac {b^4 \sqrt {a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac {b^3 \sqrt {a+b x} (5 A b-12 a B)}{768 a^2 x^2}+\frac {b^2 \sqrt {a+b x} (5 A b-12 a B)}{192 a x^3}+\frac {(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}+\frac {b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}-\frac {A (a+b x)^{7/2}}{6 a x^6} \]

[Out]

1/96*b*(5*A*b-12*B*a)*(b*x+a)^(3/2)/a/x^4+1/60*(5*A*b-12*B*a)*(b*x+a)^(5/2)/a/x^5-1/6*A*(b*x+a)^(7/2)/a/x^6+1/
512*b^5*(5*A*b-12*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(7/2)+1/192*b^2*(5*A*b-12*B*a)*(b*x+a)^(1/2)/x^3/a+1/7
68*b^3*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a^2/x^2-1/512*b^4*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a^3/x

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Rubi [A]  time = 0.10, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {78, 47, 51, 63, 208} \[ \frac {b^3 \sqrt {a+b x} (5 A b-12 a B)}{768 a^2 x^2}-\frac {b^4 \sqrt {a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac {b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}}+\frac {b^2 \sqrt {a+b x} (5 A b-12 a B)}{192 a x^3}+\frac {b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}+\frac {(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^7,x]

[Out]

(b^2*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(192*a*x^3) + (b^3*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(768*a^2*x^2) - (b^4*(
5*A*b - 12*a*B)*Sqrt[a + b*x])/(512*a^3*x) + (b*(5*A*b - 12*a*B)*(a + b*x)^(3/2))/(96*a*x^4) + ((5*A*b - 12*a*
B)*(a + b*x)^(5/2))/(60*a*x^5) - (A*(a + b*x)^(7/2))/(6*a*x^6) + (b^5*(5*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(512*a^(7/2))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin {align*} \int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx &=-\frac {A (a+b x)^{7/2}}{6 a x^6}+\frac {\left (-\frac {5 A b}{2}+6 a B\right ) \int \frac {(a+b x)^{5/2}}{x^6} \, dx}{6 a}\\ &=\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}-\frac {(b (5 A b-12 a B)) \int \frac {(a+b x)^{3/2}}{x^5} \, dx}{24 a}\\ &=\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}-\frac {\left (b^2 (5 A b-12 a B)\right ) \int \frac {\sqrt {a+b x}}{x^4} \, dx}{64 a}\\ &=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}-\frac {\left (b^3 (5 A b-12 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{384 a}\\ &=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b^3 (5 A b-12 a B) \sqrt {a+b x}}{768 a^2 x^2}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}+\frac {\left (b^4 (5 A b-12 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{512 a^2}\\ &=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b^3 (5 A b-12 a B) \sqrt {a+b x}}{768 a^2 x^2}-\frac {b^4 (5 A b-12 a B) \sqrt {a+b x}}{512 a^3 x}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}-\frac {\left (b^5 (5 A b-12 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{1024 a^3}\\ &=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b^3 (5 A b-12 a B) \sqrt {a+b x}}{768 a^2 x^2}-\frac {b^4 (5 A b-12 a B) \sqrt {a+b x}}{512 a^3 x}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}-\frac {\left (b^4 (5 A b-12 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{512 a^3}\\ &=\frac {b^2 (5 A b-12 a B) \sqrt {a+b x}}{192 a x^3}+\frac {b^3 (5 A b-12 a B) \sqrt {a+b x}}{768 a^2 x^2}-\frac {b^4 (5 A b-12 a B) \sqrt {a+b x}}{512 a^3 x}+\frac {b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac {(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac {A (a+b x)^{7/2}}{6 a x^6}+\frac {b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 58, normalized size = 0.28 \[ -\frac {(a+b x)^{7/2} \left (7 a^6 A+b^5 x^6 (5 A b-12 a B) \, _2F_1\left (\frac {7}{2},6;\frac {9}{2};\frac {b x}{a}+1\right )\right )}{42 a^7 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^7,x]

[Out]

-1/42*((a + b*x)^(7/2)*(7*a^6*A + b^5*(5*A*b - 12*a*B)*x^6*Hypergeometric2F1[7/2, 6, 9/2, 1 + (b*x)/a]))/(a^7*
x^6)

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fricas [A]  time = 0.78, size = 356, normalized size = 1.71 \[ \left [-\frac {15 \, {\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt {a} x^{6} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (1280 \, A a^{6} - 15 \, {\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \, {\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \, {\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15360 \, a^{4} x^{6}}, \frac {15 \, {\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt {-a} x^{6} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (1280 \, A a^{6} - 15 \, {\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \, {\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \, {\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{7680 \, a^{4} x^{6}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

[-1/15360*(15*(12*B*a*b^5 - 5*A*b^6)*sqrt(a)*x^6*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(1280*A*a^6
- 15*(12*B*a^2*b^4 - 5*A*a*b^5)*x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3)*x^
3 + 144*(28*B*a^5*b + 15*A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(b*x + a))/(a^4*x^6), 1/7680*(15*
(12*B*a*b^5 - 5*A*b^6)*sqrt(-a)*x^6*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (1280*A*a^6 - 15*(12*B*a^2*b^4 - 5*A*a*
b^5)*x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3)*x^3 + 144*(28*B*a^5*b + 15*A*
a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(b*x + a))/(a^4*x^6)]

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giac [A]  time = 1.72, size = 240, normalized size = 1.15 \[ \frac {\frac {15 \, {\left (12 \, B a b^{6} - 5 \, A b^{7}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {180 \, {\left (b x + a\right )}^{\frac {11}{2}} B a b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {9}{2}} B a^{2} b^{6} - 696 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{3} b^{6} + 2376 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{4} b^{6} - 1020 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{5} b^{6} + 180 \, \sqrt {b x + a} B a^{6} b^{6} - 75 \, {\left (b x + a\right )}^{\frac {11}{2}} A b^{7} + 425 \, {\left (b x + a\right )}^{\frac {9}{2}} A a b^{7} - 990 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{2} b^{7} - 990 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{3} b^{7} + 425 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{4} b^{7} - 75 \, \sqrt {b x + a} A a^{5} b^{7}}{a^{3} b^{6} x^{6}}}{7680 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^7,x, algorithm="giac")

[Out]

1/7680*(15*(12*B*a*b^6 - 5*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (180*(b*x + a)^(11/2)*B*a*b^
6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 - 696*(b*x + a)^(7/2)*B*a^3*b^6 + 2376*(b*x + a)^(5/2)*B*a^4*b^6 - 1020*(b*
x + a)^(3/2)*B*a^5*b^6 + 180*sqrt(b*x + a)*B*a^6*b^6 - 75*(b*x + a)^(11/2)*A*b^7 + 425*(b*x + a)^(9/2)*A*a*b^7
- 990*(b*x + a)^(7/2)*A*a^2*b^7 - 990*(b*x + a)^(5/2)*A*a^3*b^7 + 425*(b*x + a)^(3/2)*A*a^4*b^7 - 75*sqrt(b*x
+ a)*A*a^5*b^7)/(a^3*b^6*x^6))/b

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maple [A]  time = 0.02, size = 161, normalized size = 0.77 \[ 2 \left (\frac {\left (5 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {7}{2}}}+\frac {-\frac {\left (5 A b -12 B a \right ) \sqrt {b x +a}\, a^{2}}{1024}+\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {3}{2}} a}{3072}-\frac {\left (165 A b +116 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a}+\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{2}}-\frac {\left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{3}}+\left (-\frac {33 A b}{512}+\frac {99 B a}{640}\right ) \left (b x +a \right )^{\frac {5}{2}}}{b^{6} x^{6}}\right ) b^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^7,x)

[Out]

2*b^5*((-1/1024*(5*A*b-12*B*a)/a^3*(b*x+a)^(11/2)+17/3072/a^2*(5*A*b-12*B*a)*(b*x+a)^(9/2)-1/2560*(165*A*b+116
*B*a)/a*(b*x+a)^(7/2)+(-33/512*A*b+99/640*B*a)*(b*x+a)^(5/2)+17/3072*a*(5*A*b-12*B*a)*(b*x+a)^(3/2)-1/1024*a^2
*(5*A*b-12*B*a)*(b*x+a)^(1/2))/x^6/b^6+1/1024*(5*A*b-12*B*a)/a^(7/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

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maxima [A]  time = 1.86, size = 268, normalized size = 1.29 \[ \frac {1}{15360} \, b^{6} {\left (\frac {2 \, {\left (15 \, {\left (12 \, B a - 5 \, A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 85 \, {\left (12 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 6 \, {\left (116 \, B a^{3} + 165 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 198 \, {\left (12 \, B a^{4} - 5 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 85 \, {\left (12 \, B a^{5} - 5 \, A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (12 \, B a^{6} - 5 \, A a^{5} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{6} a^{3} b - 6 \, {\left (b x + a\right )}^{5} a^{4} b + 15 \, {\left (b x + a\right )}^{4} a^{5} b - 20 \, {\left (b x + a\right )}^{3} a^{6} b + 15 \, {\left (b x + a\right )}^{2} a^{7} b - 6 \, {\left (b x + a\right )} a^{8} b + a^{9} b} + \frac {15 \, {\left (12 \, B a - 5 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

1/15360*b^6*(2*(15*(12*B*a - 5*A*b)*(b*x + a)^(11/2) - 85*(12*B*a^2 - 5*A*a*b)*(b*x + a)^(9/2) - 6*(116*B*a^3
+ 165*A*a^2*b)*(b*x + a)^(7/2) + 198*(12*B*a^4 - 5*A*a^3*b)*(b*x + a)^(5/2) - 85*(12*B*a^5 - 5*A*a^4*b)*(b*x +
a)^(3/2) + 15*(12*B*a^6 - 5*A*a^5*b)*sqrt(b*x + a))/((b*x + a)^6*a^3*b - 6*(b*x + a)^5*a^4*b + 15*(b*x + a)^4
*a^5*b - 20*(b*x + a)^3*a^6*b + 15*(b*x + a)^2*a^7*b - 6*(b*x + a)*a^8*b + a^9*b) + 15*(12*B*a - 5*A*b)*log((s
qrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(7/2)*b))

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mupad [B]  time = 0.47, size = 254, normalized size = 1.22 \[ \frac {b^5\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-12\,B\,a\right )}{512\,a^{7/2}}-\frac {\left (\frac {33\,A\,b^6}{256}-\frac {99\,B\,a\,b^5}{320}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^6}{512}-\frac {3\,B\,a^3\,b^5}{128}\right )\,\sqrt {a+b\,x}+\left (\frac {17\,B\,a^2\,b^5}{128}-\frac {85\,A\,a\,b^6}{1536}\right )\,{\left (a+b\,x\right )}^{3/2}-\frac {17\,\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{9/2}}{1536\,a^2}+\frac {\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{11/2}}{512\,a^3}+\frac {\left (165\,A\,b^6+116\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{7/2}}{1280\,a}}{{\left (a+b\,x\right )}^6-6\,a^5\,\left (a+b\,x\right )-6\,a\,{\left (a+b\,x\right )}^5+15\,a^2\,{\left (a+b\,x\right )}^4-20\,a^3\,{\left (a+b\,x\right )}^3+15\,a^4\,{\left (a+b\,x\right )}^2+a^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^7,x)

[Out]

(b^5*atanh((a + b*x)^(1/2)/a^(1/2))*(5*A*b - 12*B*a))/(512*a^(7/2)) - (((33*A*b^6)/256 - (99*B*a*b^5)/320)*(a
+ b*x)^(5/2) + ((5*A*a^2*b^6)/512 - (3*B*a^3*b^5)/128)*(a + b*x)^(1/2) + ((17*B*a^2*b^5)/128 - (85*A*a*b^6)/15
36)*(a + b*x)^(3/2) - (17*(5*A*b^6 - 12*B*a*b^5)*(a + b*x)^(9/2))/(1536*a^2) + ((5*A*b^6 - 12*B*a*b^5)*(a + b*
x)^(11/2))/(512*a^3) + ((165*A*b^6 + 116*B*a*b^5)*(a + b*x)^(7/2))/(1280*a))/((a + b*x)^6 - 6*a^5*(a + b*x) -
6*a*(a + b*x)^5 + 15*a^2*(a + b*x)^4 - 20*a^3*(a + b*x)^3 + 15*a^4*(a + b*x)^2 + a^6)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/x**7,x)

[Out]

Timed out

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