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

Optimal. Leaf size=136 \[ \frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 B x^{9/2}}{9 b} \]

[Out]

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

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Rubi [A]  time = 0.0815894, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {80, 50, 63, 205} \[ \frac{2 a^2 x^{3/2} (A b-a B)}{3 b^4}-\frac{2 a^3 \sqrt{x} (A b-a B)}{b^5}+\frac{2 a^{7/2} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}}+\frac{2 x^{7/2} (A b-a B)}{7 b^2}-\frac{2 a x^{5/2} (A b-a B)}{5 b^3}+\frac{2 B x^{9/2}}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^(7/2)*(A + B*x))/(a + b*x),x]

[Out]

(-2*a^3*(A*b - a*B)*Sqrt[x])/b^5 + (2*a^2*(A*b - a*B)*x^(3/2))/(3*b^4) - (2*a*(A*b - a*B)*x^(5/2))/(5*b^3) + (
2*(A*b - a*B)*x^(7/2))/(7*b^2) + (2*B*x^(9/2))/(9*b) + (2*a^(7/2)*(A*b - a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]
])/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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

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

Mathematica [A]  time = 0.0962008, size = 120, normalized size = 0.88 \[ \frac{2 \sqrt{x} \left (21 a^2 b^2 x (5 A+3 B x)-105 a^3 b (3 A+B x)+315 a^4 B-9 a b^3 x^2 (7 A+5 B x)+5 b^4 x^3 (9 A+7 B x)\right )}{315 b^5}-\frac{2 a^{7/2} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(315*a^4*B - 105*a^3*b*(3*A + B*x) + 21*a^2*b^2*x*(5*A + 3*B*x) - 9*a*b^3*x^2*(7*A + 5*B*x) + 5*b^4
*x^3*(9*A + 7*B*x)))/(315*b^5) - (2*a^(7/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

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Maple [A]  time = 0.007, size = 150, normalized size = 1.1 \begin{align*}{\frac{2\,B}{9\,b}{x}^{{\frac{9}{2}}}}+{\frac{2\,A}{7\,b}{x}^{{\frac{7}{2}}}}-{\frac{2\,Ba}{7\,{b}^{2}}{x}^{{\frac{7}{2}}}}-{\frac{2\,Aa}{5\,{b}^{2}}{x}^{{\frac{5}{2}}}}+{\frac{2\,B{a}^{2}}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{2\,A{a}^{2}}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-{\frac{2\,B{a}^{3}}{3\,{b}^{4}}{x}^{{\frac{3}{2}}}}-2\,{\frac{{a}^{3}A\sqrt{x}}{{b}^{4}}}+2\,{\frac{B{a}^{4}\sqrt{x}}{{b}^{5}}}+2\,{\frac{{a}^{4}A}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) }-2\,{\frac{B{a}^{5}}{{b}^{5}\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*B*x^(9/2)/b+2/7/b*A*x^(7/2)-2/7/b^2*B*x^(7/2)*a-2/5/b^2*A*x^(5/2)*a+2/5/b^3*B*x^(5/2)*a^2+2/3/b^3*A*x^(3/2
)*a^2-2/3/b^4*B*x^(3/2)*a^3-2/b^4*A*a^3*x^(1/2)+2/b^5*B*a^4*x^(1/2)+2*a^4/b^4/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*
b)^(1/2))*A-2*a^5/b^5/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2))*B

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.36002, size = 617, normalized size = 4.54 \begin{align*} \left [-\frac{315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) - 2 \,{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}}{315 \, b^{5}}, -\frac{2 \,{\left (315 \,{\left (B a^{4} - A a^{3} b\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (35 \, B b^{4} x^{4} + 315 \, B a^{4} - 315 \, A a^{3} b - 45 \,{\left (B a b^{3} - A b^{4}\right )} x^{3} + 63 \,{\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 105 \,{\left (B a^{3} b - A a^{2} b^{2}\right )} x\right )} \sqrt{x}\right )}}{315 \, b^{5}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/315*(315*(B*a^4 - A*a^3*b)*sqrt(-a/b)*log((b*x + 2*b*sqrt(x)*sqrt(-a/b) - a)/(b*x + a)) - 2*(35*B*b^4*x^4
+ 315*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2
)*x)*sqrt(x))/b^5, -2/315*(315*(B*a^4 - A*a^3*b)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) - (35*B*b^4*x^4 + 315
*B*a^4 - 315*A*a^3*b - 45*(B*a*b^3 - A*b^4)*x^3 + 63*(B*a^2*b^2 - A*a*b^3)*x^2 - 105*(B*a^3*b - A*a^2*b^2)*x)*
sqrt(x))/b^5]

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Sympy [A]  time = 150.866, size = 313, normalized size = 2.3 \begin{align*} \begin{cases} - \frac{i A a^{\frac{7}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{5} \sqrt{\frac{1}{b}}} + \frac{i A a^{\frac{7}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{5} \sqrt{\frac{1}{b}}} - \frac{2 A a^{3} \sqrt{x}}{b^{4}} + \frac{2 A a^{2} x^{\frac{3}{2}}}{3 b^{3}} - \frac{2 A a x^{\frac{5}{2}}}{5 b^{2}} + \frac{2 A x^{\frac{7}{2}}}{7 b} + \frac{i B a^{\frac{9}{2}} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{6} \sqrt{\frac{1}{b}}} - \frac{i B a^{\frac{9}{2}} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{b^{6} \sqrt{\frac{1}{b}}} + \frac{2 B a^{4} \sqrt{x}}{b^{5}} - \frac{2 B a^{3} x^{\frac{3}{2}}}{3 b^{4}} + \frac{2 B a^{2} x^{\frac{5}{2}}}{5 b^{3}} - \frac{2 B a x^{\frac{7}{2}}}{7 b^{2}} + \frac{2 B x^{\frac{9}{2}}}{9 b} & \text{for}\: b \neq 0 \\\frac{\frac{2 A x^{\frac{9}{2}}}{9} + \frac{2 B x^{\frac{11}{2}}}{11}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I*A*a**(7/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**5*sqrt(1/b)) + I*A*a**(7/2)*log(I*sqrt(a)*sqr
t(1/b) + sqrt(x))/(b**5*sqrt(1/b)) - 2*A*a**3*sqrt(x)/b**4 + 2*A*a**2*x**(3/2)/(3*b**3) - 2*A*a*x**(5/2)/(5*b*
*2) + 2*A*x**(7/2)/(7*b) + I*B*a**(9/2)*log(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**6*sqrt(1/b)) - I*B*a**(9/2)*lo
g(I*sqrt(a)*sqrt(1/b) + sqrt(x))/(b**6*sqrt(1/b)) + 2*B*a**4*sqrt(x)/b**5 - 2*B*a**3*x**(3/2)/(3*b**4) + 2*B*a
**2*x**(5/2)/(5*b**3) - 2*B*a*x**(7/2)/(7*b**2) + 2*B*x**(9/2)/(9*b), Ne(b, 0)), ((2*A*x**(9/2)/9 + 2*B*x**(11
/2)/11)/a, True))

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Giac [A]  time = 1.17552, size = 188, normalized size = 1.38 \begin{align*} -\frac{2 \,{\left (B a^{5} - A a^{4} b\right )} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{2 \,{\left (35 \, B b^{8} x^{\frac{9}{2}} - 45 \, B a b^{7} x^{\frac{7}{2}} + 45 \, A b^{8} x^{\frac{7}{2}} + 63 \, B a^{2} b^{6} x^{\frac{5}{2}} - 63 \, A a b^{7} x^{\frac{5}{2}} - 105 \, B a^{3} b^{5} x^{\frac{3}{2}} + 105 \, A a^{2} b^{6} x^{\frac{3}{2}} + 315 \, B a^{4} b^{4} \sqrt{x} - 315 \, A a^{3} b^{5} \sqrt{x}\right )}}{315 \, b^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(B*a^5 - A*a^4*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) + 2/315*(35*B*b^8*x^(9/2) - 45*B*a*b^7*x^(7/2
) + 45*A*b^8*x^(7/2) + 63*B*a^2*b^6*x^(5/2) - 63*A*a*b^7*x^(5/2) - 105*B*a^3*b^5*x^(3/2) + 105*A*a^2*b^6*x^(3/
2) + 315*B*a^4*b^4*sqrt(x) - 315*A*a^3*b^5*sqrt(x))/b^9