3.422 \(\int \frac{(a+b x)^{5/2} (A+B x)}{x^8} \, dx\)

Optimal. Leaf size=232 \[ -\frac{5 b^4 \sqrt{a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac{b^3 \sqrt{a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac{5 b^5 \sqrt{a+b x} (A b-2 a B)}{1024 a^4 x}-\frac{5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a x^4}+\frac{b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}+\frac{(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7} \]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(384*a^2*x^3) - (5*b^4*(A*b -
2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a
+ b*x)^(3/2))/(24*a*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^7) - (5*b^
6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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Rubi [A]  time = 0.115598, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{5 b^4 \sqrt{a+b x} (A b-2 a B)}{1536 a^3 x^2}+\frac{b^3 \sqrt{a+b x} (A b-2 a B)}{384 a^2 x^3}+\frac{5 b^5 \sqrt{a+b x} (A b-2 a B)}{1024 a^4 x}-\frac{5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}+\frac{b^2 \sqrt{a+b x} (A b-2 a B)}{64 a x^4}+\frac{b (a+b x)^{3/2} (A b-2 a B)}{24 a x^5}+\frac{(a+b x)^{5/2} (A b-2 a B)}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

(b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a*x^4) + (b^3*(A*b - 2*a*B)*Sqrt[a + b*x])/(384*a^2*x^3) - (5*b^4*(A*b -
2*a*B)*Sqrt[a + b*x])/(1536*a^3*x^2) + (5*b^5*(A*b - 2*a*B)*Sqrt[a + b*x])/(1024*a^4*x) + (b*(A*b - 2*a*B)*(a
+ b*x)^(3/2))/(24*a*x^5) + ((A*b - 2*a*B)*(a + b*x)^(5/2))/(12*a*x^6) - (A*(a + b*x)^(7/2))/(7*a*x^7) - (5*b^
6*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1024*a^(9/2))

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^8} \, dx &=-\frac{A (a+b x)^{7/2}}{7 a x^7}+\frac{\left (-\frac{7 A b}{2}+7 a B\right ) \int \frac{(a+b x)^{5/2}}{x^7} \, dx}{7 a}\\ &=\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}-\frac{(5 b (A b-2 a B)) \int \frac{(a+b x)^{3/2}}{x^6} \, dx}{24 a}\\ &=\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}-\frac{\left (b^2 (A b-2 a B)\right ) \int \frac{\sqrt{a+b x}}{x^5} \, dx}{16 a}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}-\frac{\left (b^3 (A b-2 a B)\right ) \int \frac{1}{x^4 \sqrt{a+b x}} \, dx}{128 a}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b^3 (A b-2 a B) \sqrt{a+b x}}{384 a^2 x^3}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}+\frac{\left (5 b^4 (A b-2 a B)\right ) \int \frac{1}{x^3 \sqrt{a+b x}} \, dx}{768 a^2}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b^3 (A b-2 a B) \sqrt{a+b x}}{384 a^2 x^3}-\frac{5 b^4 (A b-2 a B) \sqrt{a+b x}}{1536 a^3 x^2}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}-\frac{\left (5 b^5 (A b-2 a B)\right ) \int \frac{1}{x^2 \sqrt{a+b x}} \, dx}{1024 a^3}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b^3 (A b-2 a B) \sqrt{a+b x}}{384 a^2 x^3}-\frac{5 b^4 (A b-2 a B) \sqrt{a+b x}}{1536 a^3 x^2}+\frac{5 b^5 (A b-2 a B) \sqrt{a+b x}}{1024 a^4 x}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}+\frac{\left (5 b^6 (A b-2 a B)\right ) \int \frac{1}{x \sqrt{a+b x}} \, dx}{2048 a^4}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b^3 (A b-2 a B) \sqrt{a+b x}}{384 a^2 x^3}-\frac{5 b^4 (A b-2 a B) \sqrt{a+b x}}{1536 a^3 x^2}+\frac{5 b^5 (A b-2 a B) \sqrt{a+b x}}{1024 a^4 x}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}+\frac{\left (5 b^5 (A b-2 a B)\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x}\right )}{1024 a^4}\\ &=\frac{b^2 (A b-2 a B) \sqrt{a+b x}}{64 a x^4}+\frac{b^3 (A b-2 a B) \sqrt{a+b x}}{384 a^2 x^3}-\frac{5 b^4 (A b-2 a B) \sqrt{a+b x}}{1536 a^3 x^2}+\frac{5 b^5 (A b-2 a B) \sqrt{a+b x}}{1024 a^4 x}+\frac{b (A b-2 a B) (a+b x)^{3/2}}{24 a x^5}+\frac{(A b-2 a B) (a+b x)^{5/2}}{12 a x^6}-\frac{A (a+b x)^{7/2}}{7 a x^7}-\frac{5 b^6 (A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{1024 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0228425, size = 57, normalized size = 0.25 \[ -\frac{(a+b x)^{7/2} \left (a^7 A+b^6 x^7 (2 a B-A b) \, _2F_1\left (\frac{7}{2},7;\frac{9}{2};\frac{b x}{a}+1\right )\right )}{7 a^8 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(5/2)*(A + B*x))/x^8,x]

[Out]

-((a + b*x)^(7/2)*(a^7*A + b^6*(-(A*b) + 2*a*B)*x^7*Hypergeometric2F1[7/2, 7, 9/2, 1 + (b*x)/a]))/(7*a^8*x^7)

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Maple [A]  time = 0.014, size = 169, normalized size = 0.7 \begin{align*} 2\,{b}^{6} \left ({\frac{1}{{b}^{7}{x}^{7}} \left ({\frac{ \left ( 5\,Ab-10\,Ba \right ) \left ( bx+a \right ) ^{13/2}}{2048\,{a}^{4}}}-{\frac{ \left ( 25\,Ab-50\,Ba \right ) \left ( bx+a \right ) ^{11/2}}{1536\,{a}^{3}}}+{\frac{ \left ( 283\,Ab-566\,Ba \right ) \left ( bx+a \right ) ^{9/2}}{6144\,{a}^{2}}}-1/14\,{\frac{Ab \left ( bx+a \right ) ^{7/2}}{a}}+ \left ( -{\frac{283\,Ab}{6144}}+{\frac{283\,Ba}{3072}} \right ) \left ( bx+a \right ) ^{5/2}+{\frac{25\,a \left ( Ab-2\,Ba \right ) \left ( bx+a \right ) ^{3/2}}{1536}}-{\frac{5\,{a}^{2} \left ( Ab-2\,Ba \right ) \sqrt{bx+a}}{2048}} \right ) }-{\frac{5\,Ab-10\,Ba}{2048\,{a}^{9/2}}{\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^8,x)

[Out]

2*b^6*((5/2048*(A*b-2*B*a)/a^4*(b*x+a)^(13/2)-25/1536*(A*b-2*B*a)/a^3*(b*x+a)^(11/2)+283/6144*(A*b-2*B*a)/a^2*
(b*x+a)^(9/2)-1/14*A*b/a*(b*x+a)^(7/2)+(-283/6144*A*b+283/3072*B*a)*(b*x+a)^(5/2)+25/1536*a*(A*b-2*B*a)*(b*x+a
)^(3/2)-5/2048*a^2*(A*b-2*B*a)*(b*x+a)^(1/2))/b^7/x^7-5/2048*(A*b-2*B*a)/a^(9/2)*arctanh((b*x+a)^(1/2)/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((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.46797, size = 919, normalized size = 3.96 \begin{align*} \left [-\frac{105 \,{\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt{a} x^{7} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (3072 \, A a^{7} + 105 \,{\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \,{\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \,{\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \,{\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \,{\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \,{\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt{b x + a}}{43008 \, a^{5} x^{7}}, -\frac{105 \,{\left (2 \, B a b^{6} - A b^{7}\right )} \sqrt{-a} x^{7} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (3072 \, A a^{7} + 105 \,{\left (2 \, B a^{2} b^{5} - A a b^{6}\right )} x^{6} - 70 \,{\left (2 \, B a^{3} b^{4} - A a^{2} b^{5}\right )} x^{5} + 56 \,{\left (2 \, B a^{4} b^{3} - A a^{3} b^{4}\right )} x^{4} + 48 \,{\left (126 \, B a^{5} b^{2} + A a^{4} b^{3}\right )} x^{3} + 128 \,{\left (70 \, B a^{6} b + 37 \, A a^{5} b^{2}\right )} x^{2} + 256 \,{\left (14 \, B a^{7} + 29 \, A a^{6} b\right )} x\right )} \sqrt{b x + a}}{21504 \, a^{5} x^{7}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="fricas")

[Out]

[-1/43008*(105*(2*B*a*b^6 - A*b^7)*sqrt(a)*x^7*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(3072*A*a^7 +
105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(12
6*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x +
a))/(a^5*x^7), -1/21504*(105*(2*B*a*b^6 - A*b^7)*sqrt(-a)*x^7*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (3072*A*a^7
+ 105*(2*B*a^2*b^5 - A*a*b^6)*x^6 - 70*(2*B*a^3*b^4 - A*a^2*b^5)*x^5 + 56*(2*B*a^4*b^3 - A*a^3*b^4)*x^4 + 48*(
126*B*a^5*b^2 + A*a^4*b^3)*x^3 + 128*(70*B*a^6*b + 37*A*a^5*b^2)*x^2 + 256*(14*B*a^7 + 29*A*a^6*b)*x)*sqrt(b*x
+ a))/(a^5*x^7)]

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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((b*x+a)**(5/2)*(B*x+A)/x**8,x)

[Out]

Timed out

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Giac [A]  time = 1.29551, size = 346, normalized size = 1.49 \begin{align*} -\frac{\frac{105 \,{\left (2 \, B a b^{7} - A b^{8}\right )} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{210 \,{\left (b x + a\right )}^{\frac{13}{2}} B a b^{7} - 1400 \,{\left (b x + a\right )}^{\frac{11}{2}} B a^{2} b^{7} + 3962 \,{\left (b x + a\right )}^{\frac{9}{2}} B a^{3} b^{7} - 3962 \,{\left (b x + a\right )}^{\frac{5}{2}} B a^{5} b^{7} + 1400 \,{\left (b x + a\right )}^{\frac{3}{2}} B a^{6} b^{7} - 210 \, \sqrt{b x + a} B a^{7} b^{7} - 105 \,{\left (b x + a\right )}^{\frac{13}{2}} A b^{8} + 700 \,{\left (b x + a\right )}^{\frac{11}{2}} A a b^{8} - 1981 \,{\left (b x + a\right )}^{\frac{9}{2}} A a^{2} b^{8} + 3072 \,{\left (b x + a\right )}^{\frac{7}{2}} A a^{3} b^{8} + 1981 \,{\left (b x + a\right )}^{\frac{5}{2}} A a^{4} b^{8} - 700 \,{\left (b x + a\right )}^{\frac{3}{2}} A a^{5} b^{8} + 105 \, \sqrt{b x + a} A a^{6} b^{8}}{a^{4} b^{7} x^{7}}}{21504 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2)*(B*x+A)/x^8,x, algorithm="giac")

[Out]

-1/21504*(105*(2*B*a*b^7 - A*b^8)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (210*(b*x + a)^(13/2)*B*a*b^
7 - 1400*(b*x + a)^(11/2)*B*a^2*b^7 + 3962*(b*x + a)^(9/2)*B*a^3*b^7 - 3962*(b*x + a)^(5/2)*B*a^5*b^7 + 1400*(
b*x + a)^(3/2)*B*a^6*b^7 - 210*sqrt(b*x + a)*B*a^7*b^7 - 105*(b*x + a)^(13/2)*A*b^8 + 700*(b*x + a)^(11/2)*A*a
*b^8 - 1981*(b*x + a)^(9/2)*A*a^2*b^8 + 3072*(b*x + a)^(7/2)*A*a^3*b^8 + 1981*(b*x + a)^(5/2)*A*a^4*b^8 - 700*
(b*x + a)^(3/2)*A*a^5*b^8 + 105*sqrt(b*x + a)*A*a^6*b^8)/(a^4*b^7*x^7))/b