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

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

[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))

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Rubi [A]  time = 0.088574, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {80, 50, 63, 217, 206} \[ \frac{7 a^2 x^{3/2} \sqrt{a+b x} (10 A b-9 a B)}{192 b^4}-\frac{7 a^3 \sqrt{x} \sqrt{a+b x} (10 A b-9 a B)}{128 b^5}+\frac{7 a^4 (10 A b-9 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{128 b^{11/2}}+\frac{x^{7/2} \sqrt{a+b x} (10 A b-9 a B)}{40 b^2}-\frac{7 a x^{5/2} \sqrt{a+b x} (10 A b-9 a B)}{240 b^3}+\frac{B x^{9/2} \sqrt{a+b x}}{5 b} \]

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 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 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 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]

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])

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 = 0.312337, size = 133, normalized size = 0.69 \[ \frac{\sqrt{a+b x} \left (\frac{(10 A b-9 a B) \left (b x \sqrt{\frac{b x}{a}+1} \left (70 a^2 b x-105 a^3-56 a b^2 x^2+48 b^3 x^3\right )+105 a^{7/2} \sqrt{b} \sqrt{x} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )\right )}{\sqrt{\frac{b x}{a}+1}}+384 b^5 B x^5\right )}{1920 b^6 \sqrt{x}} \]

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])

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Maple [A]  time = 0.014, size = 260, normalized size = 1.4 \begin{align*}{\frac{1}{3840}\sqrt{x}\sqrt{bx+a} \left ( 768\,B{x}^{4}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }+960\,A{x}^{3}{b}^{9/2}\sqrt{x \left ( bx+a \right ) }-864\,B{x}^{3}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }-1120\,A{x}^{2}a{b}^{7/2}\sqrt{x \left ( bx+a \right ) }+1008\,B{x}^{2}{a}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+1400\,A\sqrt{x \left ( bx+a \right ) }{b}^{5/2}x{a}^{2}-1260\,B\sqrt{x \left ( bx+a \right ) }{b}^{3/2}x{a}^{3}+1050\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{4}b-2100\,A\sqrt{x \left ( bx+a \right ) }{b}^{3/2}{a}^{3}-945\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{5}+1890\,B\sqrt{x \left ( bx+a \right ) }\sqrt{b}{a}^{4} \right ){b}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \end{align*}

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^4*b^(9/2)*(x*(b*x+a))^(1/2)+960*A*x^3*b^(9/2)*(x*(b*x+a))^(1/2)
-864*B*x^3*a*b^(7/2)*(x*(b*x+a))^(1/2)-1120*A*x^2*a*b^(7/2)*(x*(b*x+a))^(1/2)+1008*B*x^2*a^2*b^(5/2)*(x*(b*x+a
))^(1/2)+1400*A*(x*(b*x+a))^(1/2)*b^(5/2)*x*a^2-1260*B*(x*(b*x+a))^(1/2)*b^(3/2)*x*a^3+1050*A*ln(1/2*(2*(x*(b*
x+a))^(1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^4*b-2100*A*(x*(b*x+a))^(1/2)*b^(3/2)*a^3-945*B*ln(1/2*(2*(x*(b*x+a))^(
1/2)*b^(1/2)+2*b*x+a)/b^(1/2))*a^5+1890*B*(x*(b*x+a))^(1/2)*b^(1/2)*a^4)/(x*(b*x+a))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.69864, size = 740, normalized size = 3.85 \begin{align*} \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{align*}

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]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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