3.4.97 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=232 \[ -\frac {5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}}+\frac {5 b^5 \sqrt {a+b x} (A b-2 a B)}{1024 a^4 x}-\frac {5 b^4 \sqrt {a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac {b^3 \sqrt {a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac {b^2 \sqrt {a+b x} (A b-2 a B)}{64 a x^4}+\frac {(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}+\frac {b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}-\frac {A (a+b x)^{7/2}}{7 a x^7} \]

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Rubi [A]  time = 0.12, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \begin {gather*} -\frac {5 b^4 \sqrt {a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac {b^3 \sqrt {a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac {5 b^5 \sqrt {a+b x} (A b-2 a B)}{1024 a^4 x}-\frac {5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}}+\frac {b^2 \sqrt {a+b x} (A b-2 a B)}{64 a x^4}+\frac {b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}+\frac {(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(384*a^2*x^3) - (5*b^4*(A*b -
2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a
+ b*x)^(3/2))/(24*a*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^7) - (5*b^
6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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^8} \, dx &=-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (-\frac {7 A b}{2}+7 a B\right ) \int \frac {(a+b x)^{5/2}}{x^7} \, dx}{7 a}\\ &=\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {(5 b (A b-2 a B)) \int \frac {(a+b x)^{3/2}}{x^6} \, dx}{24 a}\\ &=\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (b^2 (A b-2 a B)\right ) \int \frac {\sqrt {a+b x}}{x^5} \, dx}{16 a}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (b^3 (A b-2 a B)\right ) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{128 a}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^4 (A b-2 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{768 a^2}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {\left (5 b^5 (A b-2 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{1024 a^3}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^6 (A b-2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{2048 a^4}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}+\frac {\left (5 b^5 (A b-2 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{1024 a^4}\\ &=\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a x^4}+\frac {b^3 (A b-2 a B) \sqrt {a+b x}}{384 a^2 x^3}-\frac {5 b^4 (A b-2 a B) \sqrt {a+b x}}{1536 a^3 x^2}+\frac {5 b^5 (A b-2 a B) \sqrt {a+b x}}{1024 a^4 x}+\frac {b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac {(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac {A (a+b x)^{7/2}}{7 a x^7}-\frac {5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 57, normalized size = 0.25 \begin {gather*} -\frac {(a+b x)^{7/2} \left (a^7 A+b^6 x^7 (2 a B-A b) \, _2F_1\left (\frac {7}{2},7;\frac {9}{2};\frac {b x}{a}+1\right )\right )}{7 a^8 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

-1/7*((a + b*x)^(7/2)*(a^7*A + b^6*(-(A*b) + 2*a*B)*x^7*Hypergeometric2F1[7/2, 7, 9/2, 1 + (b*x)/a]))/(a^8*x^7
)

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IntegrateAlgebraic [A]  time = 0.38, size = 214, normalized size = 0.92 \begin {gather*} \frac {5 \left (2 a b^6 B-A b^7\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1024 a^{9/2}}+\frac {\sqrt {a+b x} \left (210 a^7 B-105 a^6 A b-1400 a^6 B (a+b x)+700 a^5 A b (a+b x)+3962 a^5 B (a+b x)^2-1981 a^4 A b (a+b x)^2-3072 a^3 A b (a+b x)^3-3962 a^3 B (a+b x)^4+1981 a^2 A b (a+b x)^4+1400 a^2 B (a+b x)^5-700 a A b (a+b x)^5+105 A b (a+b x)^6-210 a B (a+b x)^6\right )}{21504 a^4 b x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

(Sqrt[a + b*x]*(-105*a^6*A*b + 210*a^7*B + 700*a^5*A*b*(a + b*x) - 1400*a^6*B*(a + b*x) - 1981*a^4*A*b*(a + b*
x)^2 + 3962*a^5*B*(a + b*x)^2 - 3072*a^3*A*b*(a + b*x)^3 + 1981*a^2*A*b*(a + b*x)^4 - 3962*a^3*B*(a + b*x)^4 -
700*a*A*b*(a + b*x)^5 + 1400*a^2*B*(a + b*x)^5 + 105*A*b*(a + b*x)^6 - 210*a*B*(a + b*x)^6))/(21504*a^4*b*x^7
) + (5*(-(A*b^7) + 2*a*b^6*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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fricas [A]  time = 1.32, size = 401, normalized size = 1.73 \begin {gather*} \left [-\frac {105 \, {\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt {a} x^{7} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (3072 \, A a^{7} + 105 \, {\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \, {\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \, {\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \, {\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \, {\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \, {\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{43008 \, a^{5} x^{7}}, -\frac {105 \, {\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt {-a} x^{7} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (3072 \, A a^{7} + 105 \, {\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \, {\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \, {\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \, {\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \, {\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \, {\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt {b x + a}}{21504 \, a^{5} x^{7}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

[-1/43008*(105*(2*B*a*b^6 - A*b^7)*sqrt(a)*x^7*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(3072*A*a^7 +
105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(12
6*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x +
a))/(a^5*x^7), -1/21504*(105*(2*B*a*b^6 - A*b^7)*sqrt(-a)*x^7*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (3072*A*a^7
+ 105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(
126*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x
+ a))/(a^5*x^7)]

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giac [A]  time = 1.47, size = 256, normalized size = 1.10 \begin {gather*} -\frac {\frac {105 \, {\left (2 \, B a b^{7} - A b^{8}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {210 \, {\left (b x + a\right )}^{\frac {13}{2}} B a b^{7} - 1400 \, {\left (b x + a\right )}^{\frac {11}{2}} B a^{2} b^{7} + 3962 \, {\left (b x + a\right )}^{\frac {9}{2}} B a^{3} b^{7} - 3962 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{5} b^{7} + 1400 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{6} b^{7} - 210 \, \sqrt {b x + a} B a^{7} b^{7} - 105 \, {\left (b x + a\right )}^{\frac {13}{2}} A b^{8} + 700 \, {\left (b x + a\right )}^{\frac {11}{2}} A a b^{8} - 1981 \, {\left (b x + a\right )}^{\frac {9}{2}} A a^{2} b^{8} + 3072 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{3} b^{8} + 1981 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{4} b^{8} - 700 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{5} b^{8} + 105 \, \sqrt {b x + a} A a^{6} b^{8}}{a^{4} b^{7} x^{7}}}{21504 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/21504*(105*(2*B*a*b^7 - A*b^8)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (210*(b*x + a)^(13/2)*B*a*b^
7 - 1400*(b*x + a)^(11/2)*B*a^2*b^7 + 3962*(b*x + a)^(9/2)*B*a^3*b^7 - 3962*(b*x + a)^(5/2)*B*a^5*b^7 + 1400*(
b*x + a)^(3/2)*B*a^6*b^7 - 210*sqrt(b*x + a)*B*a^7*b^7 - 105*(b*x + a)^(13/2)*A*b^8 + 700*(b*x + a)^(11/2)*A*a
*b^8 - 1981*(b*x + a)^(9/2)*A*a^2*b^8 + 3072*(b*x + a)^(7/2)*A*a^3*b^8 + 1981*(b*x + a)^(5/2)*A*a^4*b^8 - 700*
(b*x + a)^(3/2)*A*a^5*b^8 + 105*sqrt(b*x + a)*A*a^6*b^8)/(a^4*b^7*x^7))/b

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maple [A]  time = 0.02, size = 169, normalized size = 0.73 \begin {gather*} 2 \left (-\frac {5 \left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2048 a^{\frac {9}{2}}}+\frac {-\frac {5 \left (A b -2 B a \right ) \sqrt {b x +a}\, a^{2}}{2048}-\frac {\left (b x +a \right )^{\frac {7}{2}} A b}{14 a}+\frac {25 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {3}{2}} a}{1536}+\frac {283 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{6144 a^{2}}-\frac {25 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1536 a^{3}}+\frac {5 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {13}{2}}}{2048 a^{4}}+\left (-\frac {283 A b}{6144}+\frac {283 B a}{3072}\right ) \left (b x +a \right )^{\frac {5}{2}}}{b^{7} x^{7}}\right ) b^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(5/2)*(B*x+A)/x^8,x)

[Out]

2*b^6*((5/2048*(A*b-2*B*a)/a^4*(b*x+a)^(13/2)-25/1536*(A*b-2*B*a)/a^3*(b*x+a)^(11/2)+283/6144*(A*b-2*B*a)/a^2*
(b*x+a)^(9/2)-1/14*A*b/a*(b*x+a)^(7/2)+(-283/6144*A*b+283/3072*B*a)*(b*x+a)^(5/2)+25/1536*a*(A*b-2*B*a)*(b*x+a
)^(3/2)-5/2048*a^2*(A*b-2*B*a)*(b*x+a)^(1/2))/x^7/b^7-5/2048*(A*b-2*B*a)/a^(9/2)*arctanh((b*x+a)^(1/2)/a^(1/2)
))

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maxima [A]  time = 2.07, size = 296, normalized size = 1.28 \begin {gather*} -\frac {1}{43008} \, b^{7} {\left (\frac {2 \, {\left (3072 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{3} b + 105 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {13}{2}} - 700 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {11}{2}} + 1981 \, {\left (2 \, B a^{3} - A a^{2} b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 1981 \, {\left (2 \, B a^{5} - A a^{4} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} + 700 \, {\left (2 \, B a^{6} - A a^{5} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 105 \, {\left (2 \, B a^{7} - A a^{6} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{7} a^{4} b - 7 \, {\left (b x + a\right )}^{6} a^{5} b + 21 \, {\left (b x + a\right )}^{5} a^{6} b - 35 \, {\left (b x + a\right )}^{4} a^{7} b + 35 \, {\left (b x + a\right )}^{3} a^{8} b - 21 \, {\left (b x + a\right )}^{2} a^{9} b + 7 \, {\left (b x + a\right )} a^{10} b - a^{11} b} + \frac {105 \, {\left (2 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

-1/43008*b^7*(2*(3072*(b*x + a)^(7/2)*A*a^3*b + 105*(2*B*a - A*b)*(b*x + a)^(13/2) - 700*(2*B*a^2 - A*a*b)*(b*
x + a)^(11/2) + 1981*(2*B*a^3 - A*a^2*b)*(b*x + a)^(9/2) - 1981*(2*B*a^5 - A*a^4*b)*(b*x + a)^(5/2) + 700*(2*B
*a^6 - A*a^5*b)*(b*x + a)^(3/2) - 105*(2*B*a^7 - A*a^6*b)*sqrt(b*x + a))/((b*x + a)^7*a^4*b - 7*(b*x + a)^6*a^
5*b + 21*(b*x + a)^5*a^6*b - 35*(b*x + a)^4*a^7*b + 35*(b*x + a)^3*a^8*b - 21*(b*x + a)^2*a^9*b + 7*(b*x + a)*
a^10*b - a^11*b) + 105*(2*B*a - A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(9/2)*b))

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mupad [B]  time = 0.15, size = 279, normalized size = 1.20 \begin {gather*} \frac {\left (\frac {283\,A\,b^7}{3072}-\frac {283\,B\,a\,b^6}{1536}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^7}{1024}-\frac {5\,B\,a^3\,b^6}{512}\right )\,\sqrt {a+b\,x}+\left (\frac {25\,B\,a^2\,b^6}{384}-\frac {25\,A\,a\,b^7}{768}\right )\,{\left (a+b\,x\right )}^{3/2}-\frac {283\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{9/2}}{3072\,a^2}+\frac {25\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{11/2}}{768\,a^3}-\frac {5\,\left (A\,b^7-2\,B\,a\,b^6\right )\,{\left (a+b\,x\right )}^{13/2}}{1024\,a^4}+\frac {A\,b^7\,{\left (a+b\,x\right )}^{7/2}}{7\,a}}{7\,a\,{\left (a+b\,x\right )}^6-7\,a^6\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^7-21\,a^2\,{\left (a+b\,x\right )}^5+35\,a^3\,{\left (a+b\,x\right )}^4-35\,a^4\,{\left (a+b\,x\right )}^3+21\,a^5\,{\left (a+b\,x\right )}^2+a^7}-\frac {5\,b^6\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-2\,B\,a\right )}{1024\,a^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(5/2))/x^8,x)

[Out]

(((283*A*b^7)/3072 - (283*B*a*b^6)/1536)*(a + b*x)^(5/2) + ((5*A*a^2*b^7)/1024 - (5*B*a^3*b^6)/512)*(a + b*x)^
(1/2) + ((25*B*a^2*b^6)/384 - (25*A*a*b^7)/768)*(a + b*x)^(3/2) - (283*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(9/2))/(3
072*a^2) + (25*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(11/2))/(768*a^3) - (5*(A*b^7 - 2*B*a*b^6)*(a + b*x)^(13/2))/(102
4*a^4) + (A*b^7*(a + b*x)^(7/2))/(7*a))/(7*a*(a + b*x)^6 - 7*a^6*(a + b*x) - (a + b*x)^7 - 21*a^2*(a + b*x)^5
+ 35*a^3*(a + b*x)^4 - 35*a^4*(a + b*x)^3 + 21*a^5*(a + b*x)^2 + a^7) - (5*b^6*atanh((a + b*x)^(1/2)/a^(1/2))*
(A*b - 2*B*a))/(1024*a^(9/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2)*(B*x+A)/x**8,x)

[Out]

Timed out

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