3.526 \(\int \frac {x^{7/2} (A+B x)}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=200 \[ -\frac {35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}}+\frac {35 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-9 a B)}{64 b^5}-\frac {35 a x^{3/2} \sqrt {a+b x} (8 A b-9 a B)}{96 b^4}+\frac {7 x^{5/2} \sqrt {a+b x} (8 A b-9 a B)}{24 b^3}-\frac {x^{7/2} \sqrt {a+b x} (8 A b-9 a B)}{4 a b^2}+\frac {2 x^{9/2} (A b-a B)}{a b \sqrt {a+b x}} \]

[Out]

-35/64*a^3*(8*A*b-9*B*a)*arctanh(b^(1/2)*x^(1/2)/(b*x+a)^(1/2))/b^(11/2)+2*(A*b-B*a)*x^(9/2)/a/b/(b*x+a)^(1/2)
-35/96*a*(8*A*b-9*B*a)*x^(3/2)*(b*x+a)^(1/2)/b^4+7/24*(8*A*b-9*B*a)*x^(5/2)*(b*x+a)^(1/2)/b^3-1/4*(8*A*b-9*B*a
)*x^(7/2)*(b*x+a)^(1/2)/a/b^2+35/64*a^2*(8*A*b-9*B*a)*x^(1/2)*(b*x+a)^(1/2)/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {78, 50, 63, 217, 206} \[ \frac {35 a^2 \sqrt {x} \sqrt {a+b x} (8 A b-9 a B)}{64 b^5}-\frac {35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}}-\frac {x^{7/2} \sqrt {a+b x} (8 A b-9 a B)}{4 a b^2}+\frac {7 x^{5/2} \sqrt {a+b x} (8 A b-9 a B)}{24 b^3}-\frac {35 a x^{3/2} \sqrt {a+b x} (8 A b-9 a B)}{96 b^4}+\frac {2 x^{9/2} (A b-a B)}{a b \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(2*(A*b - a*B)*x^(9/2))/(a*b*Sqrt[a + b*x]) + (35*a^2*(8*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(64*b^5) - (35*a*
(8*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(96*b^4) + (7*(8*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(24*b^3) - ((8*A*b
- 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(4*a*b^2) - (35*a^3*(8*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])
/(64*b^(11/2))

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)^{3/2}} \, dx &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {\left (2 \left (4 A b-\frac {9 a B}{2}\right )\right ) \int \frac {x^{7/2}}{\sqrt {a+b x}} \, dx}{a b}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}+\frac {(7 (8 A b-9 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{8 b^2}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {(35 a (8 A b-9 a B)) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{48 b^3}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}+\frac {\left (35 a^2 (8 A b-9 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{64 b^4}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{128 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{64 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {\left (35 a^3 (8 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^5}\\ &=\frac {2 (A b-a B) x^{9/2}}{a b \sqrt {a+b x}}+\frac {35 a^2 (8 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{64 b^5}-\frac {35 a (8 A b-9 a B) x^{3/2} \sqrt {a+b x}}{96 b^4}+\frac {7 (8 A b-9 a B) x^{5/2} \sqrt {a+b x}}{24 b^3}-\frac {(8 A b-9 a B) x^{7/2} \sqrt {a+b x}}{4 a b^2}-\frac {35 a^3 (8 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{64 b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.26, size = 151, normalized size = 0.76 \[ \frac {\frac {(a+b x) (9 a B-8 A b) \left (105 a^{7/2} \sqrt {b} \sqrt {x} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+b x \sqrt {\frac {b x}{a}+1} \left (-105 a^3+70 a^2 b x-56 a b^2 x^2+48 b^3 x^3\right )\right )}{3 \sqrt {\frac {b x}{a}+1}}+128 b^5 x^5 (A b-a B)}{64 a b^6 \sqrt {x} \sqrt {a+b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x]

[Out]

(128*b^5*(A*b - a*B)*x^5 + ((-8*A*b + 9*a*B)*(a + b*x)*(b*x*Sqrt[1 + (b*x)/a]*(-105*a^3 + 70*a^2*b*x - 56*a*b^
2*x^2 + 48*b^3*x^3) + 105*a^(7/2)*Sqrt[b]*Sqrt[x]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]))/(3*Sqrt[1 + (b*x)/a]))/
(64*a*b^6*Sqrt[x]*Sqrt[a + b*x])

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 355, normalized size = 1.78 \[ \left [-\frac {105 \, {\left (9 \, B a^{5} - 8 \, A a^{4} b + {\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt {b} \log \left (2 \, b x - 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \, {\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \, {\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{384 \, {\left (b^{7} x + a b^{6}\right )}}, -\frac {105 \, {\left (9 \, B a^{5} - 8 \, A a^{4} b + {\left (9 \, B a^{4} b - 8 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (48 \, B b^{5} x^{4} - 945 \, B a^{4} b + 840 \, A a^{3} b^{2} - 8 \, {\left (9 \, B a b^{4} - 8 \, A b^{5}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{3} - 8 \, A a b^{4}\right )} x^{2} - 35 \, {\left (9 \, B a^{3} b^{2} - 8 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{192 \, {\left (b^{7} x + a b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="fricas")

[Out]

[-1/384*(105*(9*B*a^5 - 8*A*a^4*b + (9*B*a^4*b - 8*A*a^3*b^2)*x)*sqrt(b)*log(2*b*x - 2*sqrt(b*x + a)*sqrt(b)*s
qrt(x) + a) - 2*(48*B*b^5*x^4 - 945*B*a^4*b + 840*A*a^3*b^2 - 8*(9*B*a*b^4 - 8*A*b^5)*x^3 + 14*(9*B*a^2*b^3 -
8*A*a*b^4)*x^2 - 35*(9*B*a^3*b^2 - 8*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^7*x + a*b^6), -1/192*(105*(9*B*a^
5 - 8*A*a^4*b + (9*B*a^4*b - 8*A*a^3*b^2)*x)*sqrt(-b)*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - (48*B*b^5*x
^4 - 945*B*a^4*b + 840*A*a^3*b^2 - 8*(9*B*a*b^4 - 8*A*b^5)*x^3 + 14*(9*B*a^2*b^3 - 8*A*a*b^4)*x^2 - 35*(9*B*a^
3*b^2 - 8*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/(b^7*x + a*b^6)]

________________________________________________________________________________________

giac [A]  time = 107.66, size = 252, normalized size = 1.26 \[ \frac {1}{192} \, {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )} B {\left | b \right |}}{b^{7}} - \frac {33 \, B a b^{27} {\left | b \right |} - 8 \, A b^{28} {\left | b \right |}}{b^{34}}\right )} + \frac {315 \, B a^{2} b^{27} {\left | b \right |} - 152 \, A a b^{28} {\left | b \right |}}{b^{34}}\right )} - \frac {3 \, {\left (325 \, B a^{3} b^{27} {\left | b \right |} - 232 \, A a^{2} b^{28} {\left | b \right |}\right )}}{b^{34}}\right )} \sqrt {{\left (b x + a\right )} b - a b} \sqrt {b x + a} - \frac {35 \, {\left (9 \, B a^{4} \sqrt {b} {\left | b \right |} - 8 \, A a^{3} b^{\frac {3}{2}} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2}\right )}{128 \, b^{7}} - \frac {4 \, {\left (B a^{5} \sqrt {b} {\left | b \right |} - A a^{4} b^{\frac {3}{2}} {\left | b \right |}\right )}}{{\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="giac")

[Out]

1/192*(2*(b*x + a)*(4*(b*x + a)*(6*(b*x + a)*B*abs(b)/b^7 - (33*B*a*b^27*abs(b) - 8*A*b^28*abs(b))/b^34) + (31
5*B*a^2*b^27*abs(b) - 152*A*a*b^28*abs(b))/b^34) - 3*(325*B*a^3*b^27*abs(b) - 232*A*a^2*b^28*abs(b))/b^34)*sqr
t((b*x + a)*b - a*b)*sqrt(b*x + a) - 35/128*(9*B*a^4*sqrt(b)*abs(b) - 8*A*a^3*b^(3/2)*abs(b))*log((sqrt(b*x +
a)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2)/b^7 - 4*(B*a^5*sqrt(b)*abs(b) - A*a^4*b^(3/2)*abs(b))/(((sqrt(b*x + a
)*sqrt(b) - sqrt((b*x + a)*b - a*b))^2 + a*b)*b^6)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 330, normalized size = 1.65 \[ -\frac {\left (-96 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}-128 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}+144 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}+840 A \,a^{3} b^{2} x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-945 B \,a^{4} b x \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )+224 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}-252 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+840 A \,a^{4} b \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-945 B \,a^{5} \ln \left (\frac {2 b x +a +2 \sqrt {\left (b x +a \right ) x}\, \sqrt {b}}{2 \sqrt {b}}\right )-560 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x +630 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -1680 \sqrt {\left (b x +a \right ) x}\, A \,a^{3} b^{\frac {3}{2}}+1890 \sqrt {\left (b x +a \right ) x}\, B \,a^{4} \sqrt {b}\right ) \sqrt {x}}{384 \sqrt {\left (b x +a \right ) x}\, \sqrt {b x +a}\, b^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x)

[Out]

-1/384*(-96*((b*x+a)*x)^(1/2)*B*b^(9/2)*x^4-128*((b*x+a)*x)^(1/2)*A*b^(9/2)*x^3+144*((b*x+a)*x)^(1/2)*B*a*b^(7
/2)*x^3+224*((b*x+a)*x)^(1/2)*A*a*b^(7/2)*x^2-252*((b*x+a)*x)^(1/2)*B*a^2*b^(5/2)*x^2+840*A*ln(1/2*(2*b*x+a+2*
((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))*x*a^3*b^2-560*((b*x+a)*x)^(1/2)*A*a^2*b^(5/2)*x-945*B*ln(1/2*(2*b*x+a+2*((
b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))*x*a^4*b+630*((b*x+a)*x)^(1/2)*B*a^3*b^(3/2)*x+840*A*a^4*b*ln(1/2*(2*b*x+a+2*
((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))-1680*((b*x+a)*x)^(1/2)*A*a^3*b^(3/2)-945*B*a^5*ln(1/2*(2*b*x+a+2*((b*x+a)*
x)^(1/2)*b^(1/2))/b^(1/2))+1890*((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)^(1
/2)

________________________________________________________________________________________

maxima [A]  time = 0.96, size = 258, normalized size = 1.29 \[ \frac {B x^{5}}{4 \, \sqrt {b x^{2} + a x} b} - \frac {3 \, B a x^{4}}{8 \, \sqrt {b x^{2} + a x} b^{2}} + \frac {A x^{4}}{3 \, \sqrt {b x^{2} + a x} b} + \frac {21 \, B a^{2} x^{3}}{32 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {7 \, A a x^{3}}{12 \, \sqrt {b x^{2} + a x} b^{2}} - \frac {105 \, B a^{3} x^{2}}{64 \, \sqrt {b x^{2} + a x} b^{4}} + \frac {35 \, A a^{2} x^{2}}{24 \, \sqrt {b x^{2} + a x} b^{3}} - \frac {315 \, B a^{4} x}{64 \, \sqrt {b x^{2} + a x} b^{5}} + \frac {35 \, A a^{3} x}{8 \, \sqrt {b x^{2} + a x} b^{4}} + \frac {315 \, B a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {11}{2}}} - \frac {35 \, A a^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(3/2),x, algorithm="maxima")

[Out]

1/4*B*x^5/(sqrt(b*x^2 + a*x)*b) - 3/8*B*a*x^4/(sqrt(b*x^2 + a*x)*b^2) + 1/3*A*x^4/(sqrt(b*x^2 + a*x)*b) + 21/3
2*B*a^2*x^3/(sqrt(b*x^2 + a*x)*b^3) - 7/12*A*a*x^3/(sqrt(b*x^2 + a*x)*b^2) - 105/64*B*a^3*x^2/(sqrt(b*x^2 + a*
x)*b^4) + 35/24*A*a^2*x^2/(sqrt(b*x^2 + a*x)*b^3) - 315/64*B*a^4*x/(sqrt(b*x^2 + a*x)*b^5) + 35/8*A*a^3*x/(sqr
t(b*x^2 + a*x)*b^4) + 315/128*B*a^4*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) - 35/16*A*a^3*log(2*
b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^{7/2}\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(A + B*x))/(a + b*x)^(3/2),x)

[Out]

int((x^(7/2)*(A + B*x))/(a + b*x)^(3/2), 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)**(3/2),x)

[Out]

Timed out

________________________________________________________________________________________