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

Optimal. Leaf size=208 \[ \frac{b^3 \sqrt{a+b x} (5 A b-12 a B)}{768 a^2 x^2}-\frac{b^4 \sqrt{a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}+\frac{b^2 \sqrt{a+b x} (5 A b-12 a B)}{192 a x^3}+\frac{b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}+\frac{(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6} \]

[Out]

(b^2*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(192*a*x^3) + (b^3*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(768*a^2*x^2) - (b^4*(
5*A*b - 12*a*B)*Sqrt[a + b*x])/(512*a^3*x) + (b*(5*A*b - 12*a*B)*(a + b*x)^(3/2))/(96*a*x^4) + ((5*A*b - 12*a*
B)*(a + b*x)^(5/2))/(60*a*x^5) - (A*(a + b*x)^(7/2))/(6*a*x^6) + (b^5*(5*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(512*a^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0980468, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {78, 47, 51, 63, 208} \[ \frac{b^3 \sqrt{a+b x} (5 A b-12 a B)}{768 a^2 x^2}-\frac{b^4 \sqrt{a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}+\frac{b^2 \sqrt{a+b x} (5 A b-12 a B)}{192 a x^3}+\frac{b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}+\frac{(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(192*a*x^3) + (b^3*(5*A*b - 12*a*B)*Sqrt[a + b*x])/(768*a^2*x^2) - (b^4*(
5*A*b - 12*a*B)*Sqrt[a + b*x])/(512*a^3*x) + (b*(5*A*b - 12*a*B)*(a + b*x)^(3/2))/(96*a*x^4) + ((5*A*b - 12*a*
B)*(a + b*x)^(5/2))/(60*a*x^5) - (A*(a + b*x)^(7/2))/(6*a*x^6) + (b^5*(5*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/S
qrt[a]])/(512*a^(7/2))

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 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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 208

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

Rubi steps

\begin{align*} \int \frac{(a+b x)^{5/2} (A+B x)}{x^7} \, dx &=-\frac{A (a+b x)^{7/2}}{6 a x^6}+\frac{\left (-\frac{5 A b}{2}+6 a B\right ) \int \frac{(a+b x)^{5/2}}{x^6} \, dx}{6 a}\\ &=\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}-\frac{(b (5 A b-12 a B)) \int \frac{(a+b x)^{3/2}}{x^5} \, dx}{24 a}\\ &=\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}-\frac{\left (b^2 (5 A b-12 a B)\right ) \int \frac{\sqrt{a+b x}}{x^4} \, dx}{64 a}\\ &=\frac{b^2 (5 A b-12 a B) \sqrt{a+b x}}{192 a x^3}+\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}-\frac{\left (b^3 (5 A b-12 a B)\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{384 a}\\ &=\frac{b^2 (5 A b-12 a B) \sqrt{a+b x}}{192 a x^3}+\frac{b^3 (5 A b-12 a B) \sqrt{a+b x}}{768 a^2 x^2}+\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}+\frac{\left (b^4 (5 A b-12 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{512 a^2}\\ &=\frac{b^2 (5 A b-12 a B) \sqrt{a+b x}}{192 a x^3}+\frac{b^3 (5 A b-12 a B) \sqrt{a+b x}}{768 a^2 x^2}-\frac{b^4 (5 A b-12 a B) \sqrt{a+b x}}{512 a^3 x}+\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}-\frac{\left (b^5 (5 A b-12 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{1024 a^3}\\ &=\frac{b^2 (5 A b-12 a B) \sqrt{a+b x}}{192 a x^3}+\frac{b^3 (5 A b-12 a B) \sqrt{a+b x}}{768 a^2 x^2}-\frac{b^4 (5 A b-12 a B) \sqrt{a+b x}}{512 a^3 x}+\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}-\frac{\left (b^4 (5 A b-12 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{512 a^3}\\ &=\frac{b^2 (5 A b-12 a B) \sqrt{a+b x}}{192 a x^3}+\frac{b^3 (5 A b-12 a B) \sqrt{a+b x}}{768 a^2 x^2}-\frac{b^4 (5 A b-12 a B) \sqrt{a+b x}}{512 a^3 x}+\frac{b (5 A b-12 a B) (a+b x)^{3/2}}{96 a x^4}+\frac{(5 A b-12 a B) (a+b x)^{5/2}}{60 a x^5}-\frac{A (a+b x)^{7/2}}{6 a x^6}+\frac{b^5 (5 A b-12 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{512 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.0240761, size = 58, normalized size = 0.28 \[ -\frac{(a+b x)^{7/2} \left (7 a^6 A+b^5 x^6 (5 A b-12 a B) \, _2F_1\left (\frac{7}{2},6;\frac{9}{2};\frac{b x}{a}+1\right )\right )}{42 a^7 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*x)^(7/2)*(7*a^6*A + b^5*(5*A*b - 12*a*B)*x^6*Hypergeometric2F1[7/2, 6, 9/2, 1 + (b*x)/a]))/(42*a^7*x^
6)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 161, normalized size = 0.8 \begin{align*} 2\,{b}^{5} \left ({\frac{1}{{b}^{6}{x}^{6}} \left ( -{\frac{ \left ( 5\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1024\,{a}^{3}}}+{\frac{ \left ( 85\,Ab-204\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{3072\,{a}^{2}}}-{\frac{ \left ( 165\,Ab+116\,Ba \right ) \left ( bx+a \right ) ^{7/2}}{2560\,a}}+ \left ( -{\frac{33\,Ab}{512}}+{\frac{99\,Ba}{640}} \right ) \left ( bx+a \right ) ^{5/2}+{\frac{17\,a \left ( 5\,Ab-12\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{3072}}-{\frac{{a}^{2} \left ( 5\,Ab-12\,Ba \right ) \sqrt{bx+a}}{1024}} \right ) }+{\frac{5\,Ab-12\,Ba}{1024\,{a}^{7/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b^5*((-1/1024*(5*A*b-12*B*a)/a^3*(b*x+a)^(11/2)+17/3072/a^2*(5*A*b-12*B*a)*(b*x+a)^(9/2)-1/2560*(165*A*b+116
*B*a)/a*(b*x+a)^(7/2)+(-33/512*A*b+99/640*B*a)*(b*x+a)^(5/2)+17/3072*a*(5*A*b-12*B*a)*(b*x+a)^(3/2)-1/1024*a^2
*(5*A*b-12*B*a)*(b*x+a)^(1/2))/b^6/x^6+1/1024*(5*A*b-12*B*a)/a^(7/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.58425, size = 844, normalized size = 4.06 \begin{align*} \left [-\frac{15 \,{\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt{a} x^{6} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (1280 \, A a^{6} - 15 \,{\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \,{\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \,{\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \,{\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{15360 \, a^{4} x^{6}}, \frac{15 \,{\left (12 \, B a b^{5} - 5 \, A b^{6}\right )} \sqrt{-a} x^{6} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) -{\left (1280 \, A a^{6} - 15 \,{\left (12 \, B a^{2} b^{4} - 5 \, A a b^{5}\right )} x^{5} + 10 \,{\left (12 \, B a^{3} b^{3} - 5 \, A a^{2} b^{4}\right )} x^{4} + 8 \,{\left (372 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3}\right )} x^{3} + 144 \,{\left (28 \, B a^{5} b + 15 \, A a^{4} b^{2}\right )} x^{2} + 128 \,{\left (12 \, B a^{6} + 25 \, A a^{5} b\right )} x\right )} \sqrt{b x + a}}{7680 \, a^{4} x^{6}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/15360*(15*(12*B*a*b^5 - 5*A*b^6)*sqrt(a)*x^6*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(1280*A*a^6
- 15*(12*B*a^2*b^4 - 5*A*a*b^5)*x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3)*x^
3 + 144*(28*B*a^5*b + 15*A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(b*x + a))/(a^4*x^6), 1/7680*(15*
(12*B*a*b^5 - 5*A*b^6)*sqrt(-a)*x^6*arctan(sqrt(b*x + a)*sqrt(-a)/a) - (1280*A*a^6 - 15*(12*B*a^2*b^4 - 5*A*a*
b^5)*x^5 + 10*(12*B*a^3*b^3 - 5*A*a^2*b^4)*x^4 + 8*(372*B*a^4*b^2 + 5*A*a^3*b^3)*x^3 + 144*(28*B*a^5*b + 15*A*
a^4*b^2)*x^2 + 128*(12*B*a^6 + 25*A*a^5*b)*x)*sqrt(b*x + a))/(a^4*x^6)]

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.20173, size = 324, normalized size = 1.56 \begin{align*} \frac{\frac{15 \,{\left (12 \, B a b^{6} - 5 \, A b^{7}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{180 \,{\left (b x + a\right )}^{\frac{11}{2}} B a b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{2} b^{6} - 696 \,{\left (b x + a\right )}^{\frac{7}{2}} B a^{3} b^{6} + 2376 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{4} b^{6} - 1020 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{5} b^{6} + 180 \, \sqrt{b x + a} B a^{6} b^{6} - 75 \,{\left (b x + a\right )}^{\frac{11}{2}} A b^{7} + 425 \,{\left (b x + a\right )}^{\frac{9}{2}} A a b^{7} - 990 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{2} b^{7} - 990 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{3} b^{7} + 425 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{4} b^{7} - 75 \, \sqrt{b x + a} A a^{5} b^{7}}{a^{3} b^{6} x^{6}}}{7680 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7680*(15*(12*B*a*b^6 - 5*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (180*(b*x + a)^(11/2)*B*a*b^
6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 - 696*(b*x + a)^(7/2)*B*a^3*b^6 + 2376*(b*x + a)^(5/2)*B*a^4*b^6 - 1020*(b*
x + a)^(3/2)*B*a^5*b^6 + 180*sqrt(b*x + a)*B*a^6*b^6 - 75*(b*x + a)^(11/2)*A*b^7 + 425*(b*x + a)^(9/2)*A*a*b^7
- 990*(b*x + a)^(7/2)*A*a^2*b^7 - 990*(b*x + a)^(5/2)*A*a^3*b^7 + 425*(b*x + a)^(3/2)*A*a^4*b^7 - 75*sqrt(b*x
+ a)*A*a^5*b^7)/(a^3*b^6*x^6))/b