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

Optimal. Leaf size=192 \[ \frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}-\frac {7 a^3 \sqrt {x} \sqrt {a+b x} (10 A b-9 a B)}{128 b^5}+\frac {7 a^2 x^{3/2} \sqrt {a+b x} (10 A b-9 a B)}{192 b^4}-\frac {7 a x^{5/2} \sqrt {a+b x} (10 A b-9 a B)}{240 b^3}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-9 a B)}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b} \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 192, 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 = {80, 50, 63, 217, 206} \begin {gather*} \frac {7 a^2 x^{3/2} \sqrt {a+b x} (10 A b-9 a B)}{192 b^4}-\frac {7 a^3 \sqrt {x} \sqrt {a+b x} (10 A b-9 a B)}{128 b^5}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}+\frac {x^{7/2} \sqrt {a+b x} (10 A b-9 a B)}{40 b^2}-\frac {7 a x^{5/2} \sqrt {a+b x} (10 A b-9 a B)}{240 b^3}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*a^3*(10*A*b - 9*a*B)*Sqrt[x]*Sqrt[a + b*x])/(128*b^5) + (7*a^2*(10*A*b - 9*a*B)*x^(3/2)*Sqrt[a + b*x])/(19
2*b^4) - (7*a*(10*A*b - 9*a*B)*x^(5/2)*Sqrt[a + b*x])/(240*b^3) + ((10*A*b - 9*a*B)*x^(7/2)*Sqrt[a + b*x])/(40
*b^2) + (B*x^(9/2)*Sqrt[a + b*x])/(5*b) + (7*a^4*(10*A*b - 9*a*B)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(1
28*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 80

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

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)}{\sqrt {a+b x}} \, dx &=\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (5 A b-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{\sqrt {a+b x}} \, dx}{5 b}\\ &=\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {(7 a (10 A b-9 a B)) \int \frac {x^{5/2}}{\sqrt {a+b x}} \, dx}{80 b^2}\\ &=-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^2 (10 A b-9 a B)\right ) \int \frac {x^{3/2}}{\sqrt {a+b x}} \, dx}{96 b^3}\\ &=\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}-\frac {\left (7 a^3 (10 A b-9 a B)\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x}} \, dx}{128 b^4}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx}{256 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 A b-9 a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {\left (7 a^4 (10 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 )}{128 b^5}\\ &=-\frac {7 a^3 (10 A b-9 a B) \sqrt {x} \sqrt {a+b x}}{128 b^5}+\frac {7 a^2 (10 A b-9 a B) x^{3/2} \sqrt {a+b x}}{192 b^4}-\frac {7 a (10 A b-9 a B) x^{5/2} \sqrt {a+b x}}{240 b^3}+\frac {(10 A b-9 a B) x^{7/2} \sqrt {a+b x}}{40 b^2}+\frac {B x^{9/2} \sqrt {a+b x}}{5 b}+\frac {7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )}{128 b^{11/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 133, normalized size = 0.69 \begin {gather*} \frac {\sqrt {a+b x} \left (\frac {(10 A b-9 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 )}{\sqrt {\frac {b x}{a}+1}}+384 b^5 B x^5\right )}{1920 b^6 \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(384*b^5*B*x^5 + ((10*A*b - 9*a*B)*(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]]))/Sqrt[1 + (b*x)/a]))/(1920*b^
6*Sqrt[x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.31, size = 173, normalized size = 0.90 \begin {gather*} \frac {7 \left (9 a^5 B-10 a^4 A b\right ) \log \left (\sqrt {a+b x}-\sqrt {b} \sqrt {x}\right )}{128 b^{11/2}}+\frac {\sqrt {a+b x} \left (945 a^4 B \sqrt {x}-1050 a^3 A b \sqrt {x}-630 a^3 b B x^{3/2}+700 a^2 A b^2 x^{3/2}+504 a^2 b^2 B x^{5/2}-560 a A b^3 x^{5/2}-432 a b^3 B x^{7/2}+480 A b^4 x^{7/2}+384 b^4 B x^{9/2}\right )}{1920 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[a + b*x]*(-1050*a^3*A*b*Sqrt[x] + 945*a^4*B*Sqrt[x] + 700*a^2*A*b^2*x^(3/2) - 630*a^3*b*B*x^(3/2) - 560*
a*A*b^3*x^(5/2) + 504*a^2*b^2*B*x^(5/2) + 480*A*b^4*x^(7/2) - 432*a*b^3*B*x^(7/2) + 384*b^4*B*x^(9/2)))/(1920*
b^5) + (7*(-10*a^4*A*b + 9*a^5*B)*Log[-(Sqrt[b]*Sqrt[x]) + Sqrt[a + b*x]])/(128*b^(11/2))

________________________________________________________________________________________

fricas [A]  time = 1.71, size = 296, normalized size = 1.54 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {b} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) - 2 \, {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{3840 \, b^{6}}, \frac {105 \, {\left (9 \, B a^{5} - 10 \, A a^{4} b\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) + {\left (384 \, B b^{5} x^{4} + 945 \, B a^{4} b - 1050 \, A a^{3} b^{2} - 48 \, {\left (9 \, B a b^{4} - 10 \, A b^{5}\right )} x^{3} + 56 \, {\left (9 \, B a^{2} b^{3} - 10 \, A a b^{4}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b^{2} - 10 \, A a^{2} b^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{1920 \, b^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[-1/3840*(105*(9*B*a^5 - 10*A*a^4*b)*sqrt(b)*log(2*b*x + 2*sqrt(b*x + a)*sqrt(b)*sqrt(x) + a) - 2*(384*B*b^5*x
^4 + 945*B*a^4*b - 1050*A*a^3*b^2 - 48*(9*B*a*b^4 - 10*A*b^5)*x^3 + 56*(9*B*a^2*b^3 - 10*A*a*b^4)*x^2 - 70*(9*
B*a^3*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^6, 1/1920*(105*(9*B*a^5 - 10*A*a^4*b)*sqrt(-b)*arctan(sq
rt(b*x + a)*sqrt(-b)/(b*sqrt(x))) + (384*B*b^5*x^4 + 945*B*a^4*b - 1050*A*a^3*b^2 - 48*(9*B*a*b^4 - 10*A*b^5)*
x^3 + 56*(9*B*a^2*b^3 - 10*A*a*b^4)*x^2 - 70*(9*B*a^3*b^2 - 10*A*a^2*b^3)*x)*sqrt(b*x + a)*sqrt(x))/b^6]

________________________________________________________________________________________

giac [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(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.02, size = 260, normalized size = 1.35 \begin {gather*} \frac {\sqrt {b x +a}\, \left (768 \sqrt {\left (b x +a \right ) x}\, B \,b^{\frac {9}{2}} x^{4}+960 \sqrt {\left (b x +a \right ) x}\, A \,b^{\frac {9}{2}} x^{3}-864 \sqrt {\left (b x +a \right ) x}\, B a \,b^{\frac {7}{2}} x^{3}-1120 \sqrt {\left (b x +a \right ) x}\, A a \,b^{\frac {7}{2}} x^{2}+1008 \sqrt {\left (b x +a \right ) x}\, B \,a^{2} b^{\frac {5}{2}} x^{2}+1050 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 )+1400 \sqrt {\left (b x +a \right ) x}\, A \,a^{2} b^{\frac {5}{2}} x -1260 \sqrt {\left (b x +a \right ) x}\, B \,a^{3} b^{\frac {3}{2}} x -2100 \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}}{3840 \sqrt {\left (b x +a \right ) x}\, b^{\frac {11}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x)

[Out]

1/3840*x^(1/2)*(b*x+a)^(1/2)/b^(11/2)*(768*((b*x+a)*x)^(1/2)*B*b^(9/2)*x^4+960*((b*x+a)*x)^(1/2)*A*b^(9/2)*x^3
-864*((b*x+a)*x)^(1/2)*B*a*b^(7/2)*x^3-1120*((b*x+a)*x)^(1/2)*A*a*b^(7/2)*x^2+1008*((b*x+a)*x)^(1/2)*B*a^2*b^(
5/2)*x^2+1400*((b*x+a)*x)^(1/2)*A*a^2*b^(5/2)*x-1260*((b*x+a)*x)^(1/2)*B*a^3*b^(3/2)*x+1050*A*a^4*b*ln(1/2*(2*
b*x+a+2*((b*x+a)*x)^(1/2)*b^(1/2))/b^(1/2))-2100*((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*x+a)*x)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.94, size = 252, normalized size = 1.31 \begin {gather*} \frac {\sqrt {b x^{2} + a x} B x^{4}}{5 \, b} - \frac {9 \, \sqrt {b x^{2} + a x} B a x^{3}}{40 \, b^{2}} + \frac {\sqrt {b x^{2} + a x} A x^{3}}{4 \, b} + \frac {21 \, \sqrt {b x^{2} + a x} B a^{2} x^{2}}{80 \, b^{3}} - \frac {7 \, \sqrt {b x^{2} + a x} A a x^{2}}{24 \, b^{2}} - \frac {21 \, \sqrt {b x^{2} + a x} B a^{3} x}{64 \, b^{4}} + \frac {35 \, \sqrt {b x^{2} + a x} A a^{2} x}{96 \, b^{3}} - \frac {63 \, B a^{5} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {11}{2}}} + \frac {35 \, A a^{4} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} + \frac {63 \, \sqrt {b x^{2} + a x} B a^{4}}{128 \, b^{5}} - \frac {35 \, \sqrt {b x^{2} + a x} A a^{3}}{64 \, b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(7/2)*(B*x+A)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

1/5*sqrt(b*x^2 + a*x)*B*x^4/b - 9/40*sqrt(b*x^2 + a*x)*B*a*x^3/b^2 + 1/4*sqrt(b*x^2 + a*x)*A*x^3/b + 21/80*sqr
t(b*x^2 + a*x)*B*a^2*x^2/b^3 - 7/24*sqrt(b*x^2 + a*x)*A*a*x^2/b^2 - 21/64*sqrt(b*x^2 + a*x)*B*a^3*x/b^4 + 35/9
6*sqrt(b*x^2 + a*x)*A*a^2*x/b^3 - 63/256*B*a^5*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 35/128*
A*a^4*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) + 63/128*sqrt(b*x^2 + a*x)*B*a^4/b^5 - 35/64*sqrt(b
*x^2 + a*x)*A*a^3/b^4

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{7/2}\,\left (A+B\,x\right )}{\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^(7/2)*(A + B*x))/(a + b*x)^(1/2),x)

[Out]

int((x^(7/2)*(A + B*x))/(a + b*x)^(1/2), x)

________________________________________________________________________________________

sympy [A]  time = 103.15, size = 360, normalized size = 1.88 \begin {gather*} - \frac {35 A a^{\frac {7}{2}} \sqrt {x}}{64 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {35 A a^{\frac {5}{2}} x^{\frac {3}{2}}}{192 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {7 A a^{\frac {3}{2}} x^{\frac {5}{2}}}{96 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {A \sqrt {a} x^{\frac {7}{2}}}{24 b \sqrt {1 + \frac {b x}{a}}} + \frac {35 A a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{64 b^{\frac {9}{2}}} + \frac {A x^{\frac {9}{2}}}{4 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} + \frac {63 B a^{\frac {9}{2}} \sqrt {x}}{128 b^{5} \sqrt {1 + \frac {b x}{a}}} + \frac {21 B a^{\frac {7}{2}} x^{\frac {3}{2}}}{128 b^{4} \sqrt {1 + \frac {b x}{a}}} - \frac {21 B a^{\frac {5}{2}} x^{\frac {5}{2}}}{320 b^{3} \sqrt {1 + \frac {b x}{a}}} + \frac {3 B a^{\frac {3}{2}} x^{\frac {7}{2}}}{80 b^{2} \sqrt {1 + \frac {b x}{a}}} - \frac {B \sqrt {a} x^{\frac {9}{2}}}{40 b \sqrt {1 + \frac {b x}{a}}} - \frac {63 B a^{5} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{128 b^{\frac {11}{2}}} + \frac {B x^{\frac {11}{2}}}{5 \sqrt {a} \sqrt {1 + \frac {b x}{a}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)/(b*x+a)**(1/2),x)

[Out]

-35*A*a**(7/2)*sqrt(x)/(64*b**4*sqrt(1 + b*x/a)) - 35*A*a**(5/2)*x**(3/2)/(192*b**3*sqrt(1 + b*x/a)) + 7*A*a**
(3/2)*x**(5/2)/(96*b**2*sqrt(1 + b*x/a)) - A*sqrt(a)*x**(7/2)/(24*b*sqrt(1 + b*x/a)) + 35*A*a**4*asinh(sqrt(b)
*sqrt(x)/sqrt(a))/(64*b**(9/2)) + A*x**(9/2)/(4*sqrt(a)*sqrt(1 + b*x/a)) + 63*B*a**(9/2)*sqrt(x)/(128*b**5*sqr
t(1 + b*x/a)) + 21*B*a**(7/2)*x**(3/2)/(128*b**4*sqrt(1 + b*x/a)) - 21*B*a**(5/2)*x**(5/2)/(320*b**3*sqrt(1 +
b*x/a)) + 3*B*a**(3/2)*x**(7/2)/(80*b**2*sqrt(1 + b*x/a)) - B*sqrt(a)*x**(9/2)/(40*b*sqrt(1 + b*x/a)) - 63*B*a
**5*asinh(sqrt(b)*sqrt(x)/sqrt(a))/(128*b**(11/2)) + B*x**(11/2)/(5*sqrt(a)*sqrt(1 + b*x/a))

________________________________________________________________________________________