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

Optimal. Leaf size=200 \[ \frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}-\frac {35 a \sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{8 b^5}+\frac {35 x^{3/2} \sqrt {a+b x} (2 A b-3 a B)}{12 b^4}-\frac {7 x^{5/2} \sqrt {a+b x} (2 A b-3 a B)}{3 a b^3}+\frac {2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt {a+b x}}+\frac {2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

[Out]

2/3*(A*b-B*a)*x^(9/2)/a/b/(b*x+a)^(3/2)+35/8*a^2*(2*A*b-3*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(11/2)
+2*(2*A*b-3*B*a)*x^(7/2)/a/b^2/(b*x+a)^(1/2)+35/12*(2*A*b-3*B*a)*x^(3/2)*(b*x+a)^(1/2)/b^4-7/3*(2*A*b-3*B*a)*x
^(5/2)*(b*x+a)^(1/2)/a/b^3-35/8*a*(2*A*b-3*B*a)*x^(1/2)*(b*x+a)^(1/2)/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {78, 47, 50, 63, 217, 206} \[ \frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}+\frac {2 x^{7/2} (2 A b-3 a B)}{a b^2 \sqrt {a+b x}}-\frac {7 x^{5/2} \sqrt {a+b x} (2 A b-3 a B)}{3 a b^3}+\frac {35 x^{3/2} \sqrt {a+b x} (2 A b-3 a B)}{12 b^4}-\frac {35 a \sqrt {x} \sqrt {a+b x} (2 A b-3 a B)}{8 b^5}+\frac {2 x^{9/2} (A b-a B)}{3 a b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(A*b - a*B)*x^(9/2))/(3*a*b*(a + b*x)^(3/2)) + (2*(2*A*b - 3*a*B)*x^(7/2))/(a*b^2*Sqrt[a + b*x]) - (35*a*(2
*A*b - 3*a*B)*Sqrt[x]*Sqrt[a + b*x])/(8*b^5) + (35*(2*A*b - 3*a*B)*x^(3/2)*Sqrt[a + b*x])/(12*b^4) - (7*(2*A*b
- 3*a*B)*x^(5/2)*Sqrt[a + b*x])/(3*a*b^3) + (35*a^2*(2*A*b - 3*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])
/(8*b^(11/2))

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{(a+b x)^{5/2}} \, dx &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}-\frac {\left (2 \left (3 A b-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{(a+b x)^{3/2}} \, dx}{3 a b}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {(7 (2 A b-3 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{a b^2}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {(35 (2 A b-3 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}-\frac {(35 a (2 A b-3 a B)) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{8 b^4}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{16 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {\left (35 a^2 (2 A b-3 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{3 a b (a+b x)^{3/2}}+\frac {2 (2 A b-3 a B) x^{7/2}}{a b^2 \sqrt {a+b x}}-\frac {35 a (2 A b-3 a B) \sqrt {x} \sqrt {a+b x}}{8 b^5}+\frac {35 (2 A b-3 a B) x^{3/2} \sqrt {a+b x}}{12 b^4}-\frac {7 (2 A b-3 a B) x^{5/2} \sqrt {a+b x}}{3 a b^3}+\frac {35 a^2 (2 A b-3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{8 b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.07, size = 80, normalized size = 0.40 \[ \frac {2 x^{9/2} \left ((a+b x) \sqrt {\frac {b x}{a}+1} (3 a B-2 A b) \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {b x}{a}\right )+3 a (A b-a B)\right )}{9 a^2 b (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(9/2)*(3*a*(A*b - a*B) + (-2*A*b + 3*a*B)*(a + b*x)*Sqrt[1 + (b*x)/a]*Hypergeometric2F1[3/2, 9/2, 11/2, -
((b*x)/a)]))/(9*a^2*b*(a + b*x)^(3/2))

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 424, normalized size = 2.12 \[ \left [-\frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{48 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}, \frac {105 \, {\left (3 \, B a^{5} - 2 \, A a^{4} b + {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x^{2} + 2 \, {\left (3 \, B a^{4} b - 2 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (8 \, B b^{5} x^{4} + 315 \, B a^{4} b - 210 \, A a^{3} b^{2} - 6 \, {\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} x^{3} + 21 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} x^{2} + 140 \, {\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, {\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/48*(105*(3*B*a^5 - 2*A*a^4*b + (3*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 2*(3*B*a^4*b - 2*A*a^3*b^2)*x)*sqrt(b)*lo
g(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(8*B*b^5*x^4 + 315*B*a^4*b - 210*A*a^3*b^2 - 6*(3*B*a*b^4 -
2*A*b^5)*x^3 + 21*(3*B*a^2*b^3 - 2*A*a*b^4)*x^2 + 140*(3*B*a^3*b^2 - 2*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(
b^8*x^2 + 2*a*b^7*x + a^2*b^6), 1/24*(105*(3*B*a^5 - 2*A*a^4*b + (3*B*a^3*b^2 - 2*A*a^2*b^3)*x^2 + 2*(3*B*a^4*
b - 2*A*a^3*b^2)*x)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (8*B*b^5*x^4 + 315*B*a^4*b - 210*A*a
^3*b^2 - 6*(3*B*a*b^4 - 2*A*b^5)*x^3 + 21*(3*B*a^2*b^3 - 2*A*a*b^4)*x^2 + 140*(3*B*a^3*b^2 - 2*A*a^2*b^3)*x)*s
qrt(b*x + a)*sqrt(x))/(b^8*x^2 + 2*a*b^7*x + a^2*b^6)]

________________________________________________________________________________________

giac [B]  time = 98.12, size = 382, normalized size = 1.91 \[ \frac {1}{24} \, \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {25 \, B a b^{20} {\left | b \right |} - 6 \, A b^{21} {\left | b \right |}}{b^{27}}\right )} + \frac {3 \, {\left (55 \, B a^{2} b^{20} {\left | b \right |} - 26 \, A a b^{21} {\left | b \right |}\right )}}{b^{27}}\right )} + \frac {35 \, {\left (3 \, B a^{3} \sqrt {b} {\left | b \right |} - 2 \, A a^{2} b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{16 \, b^{7}} + \frac {4 \, {\left (15 \, B a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} \sqrt {b} {\left | b \right |} + 24 \, B a^{5} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {3}{2}} {\left | b \right |} - 12 \, A a^{3} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{4} b^{\frac {3}{2}} {\left | b \right |} + 13 \, B a^{6} b^{\frac {5}{2}} {\left | b \right |} - 18 \, A a^{4} {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} b^{\frac {5}{2}} {\left | b \right |} - 10 \, A a^{5} b^{\frac {7}{2}} {\left | b \right |}\right )}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*sqrt((b*x + a)*b - a*b)*sqrt(b*x + a)*(2*(b*x + a)*(4*(b*x + a)*B*abs(b)/b^7 - (25*B*a*b^20*abs(b) - 6*A*
b^21*abs(b))/b^27) + 3*(55*B*a^2*b^20*abs(b) - 26*A*a*b^21*abs(b))/b^27) + 35/16*(3*B*a^3*sqrt(b)*abs(b) - 2*A
*a^2*b^(3/2)*abs(b))*log((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^7 + 4/3*(15*B*a^4*(sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*sqrt(b)*abs(b) + 24*B*a^5*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b -
a*b))^2*b^(3/2)*abs(b) - 12*A*a^3*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^4*b^(3/2)*abs(b) + 13*B*a^
6*b^(5/2)*abs(b) - 18*A*a^4*(sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2*b^(5/2)*abs(b) - 10*A*a^5*b^(7
/2)*abs(b))/(((sqrt(b*x + a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)^3*b^6)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 406, normalized size = 2.03 \[ \frac {\left (16 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}+210 A \,a^{2} b^{3} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-315 B \,a^{3} b^{2} x^{2} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+24 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}-36 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}+420 A \,a^{3} b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-630 B \,a^{4} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-84 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}+126 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+210 A \,a^{4} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-315 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-560 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x +840 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -420 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+630 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} \sqrt {b}\right ) \sqrt {x}}{48 \sqrt {\left (b x +a \right ) x}\, \left (b x +a \right )^{\frac {3}{2}} b^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(16*((b*x+a)*x)^(1/2)*B*b^(9/2)*x^4+24*((b*x+a)*x)^(1/2)*A*b^(9/2)*x^3-36*((b*x+a)*x)^(1/2)*B*a*b^(7/2)*x
^3+210*A*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))*x^2*a^2*b^3-84*((b*x+a)*x)^(1/2)*A*a*b^(7/2)*x^
2-315*B*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))*x^2*a^3*b^2+126*((b*x+a)*x)^(1/2)*B*a^2*b^(5/2)*
x^2+420*A*a^3*b^2*x*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))-560*((b*x+a)*x)^(1/2)*A*a^2*b^(5/2)*
x-630*B*a^4*b*x*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))+840*((b*x+a)*x)^(1/2)*B*a^3*b^(3/2)*x+21
0*A*a^4*b*ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))-420*((b*x+a)*x)^(1/2)*A*a^3*b^(3/2)-315*B*a^5*
ln(1/2*(2*b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))+630*((b*x+a)*x)^(1/2)*B*a^4*b^(1/2))/b^(11/2)*x^(1/2)/((
b*x+a)*x)^(1/2)/(b*x+a)^(3/2)

________________________________________________________________________________________

maxima [B]  time = 0.98, size = 416, normalized size = 2.08 \[ \frac {B x^{6}}{3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} - \frac {3 \, B a x^{5}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {A x^{5}}{2 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {35 \, B a^{3} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{16 \, b^{3}} - \frac {35 \, A a^{2} x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b} + \frac {a x}{{\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} - \frac {2 \, x}{\sqrt {b x^{2} + a x} a b} - \frac {1}{\sqrt {b x^{2} + a x} b^{2}}\right )}}{24 \, b^{2}} + \frac {21 \, B a^{2} x^{4}}{8 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{3}} - \frac {7 \, A a x^{4}}{4 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} b^{2}} + \frac {35 \, B a^{3} x}{4 \, \sqrt {b x^{2} + a x} b^{5}} - \frac {35 \, A a^{2} x}{6 \, \sqrt {b x^{2} + a x} b^{4}} - \frac {105 \, B a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{8 \, b^{\frac {9}{2}}} + \frac {35 \, \sqrt {b x^{2} + a x} B a^{2}}{8 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*x^6/((b*x^2 + a*x)^(3/2)*b) - 3/4*B*a*x^5/((b*x^2 + a*x)^(3/2)*b^2) + 1/2*A*x^5/((b*x^2 + a*x)^(3/2)*b)
+ 35/16*B*a^3*x*(3*x^2/((b*x^2 + a*x)^(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2) - 2*x/(sqrt(b*x^2 + a*x)*a*b) -
1/(sqrt(b*x^2 + a*x)*b^2))/b^3 - 35/24*A*a^2*x*(3*x^2/((b*x^2 + a*x)^(3/2)*b) + a*x/((b*x^2 + a*x)^(3/2)*b^2)
- 2*x/(sqrt(b*x^2 + a*x)*a*b) - 1/(sqrt(b*x^2 + a*x)*b^2))/b^2 + 21/8*B*a^2*x^4/((b*x^2 + a*x)^(3/2)*b^3) - 7
/4*A*a*x^4/((b*x^2 + a*x)^(3/2)*b^2) + 35/4*B*a^3*x/(sqrt(b*x^2 + a*x)*b^5) - 35/6*A*a^2*x/(sqrt(b*x^2 + a*x)*
b^4) - 105/16*B*a^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 35/8*A*a^2*log(2*b*x + a + 2*sqrt(
b*x^2 + a*x)*sqrt(b))/b^(9/2) + 35/8*sqrt(b*x^2 + a*x)*B*a^2/b^5 - 35/12*sqrt(b*x^2 + a*x)*A*a/b^4

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________