3.5.8 \(\int \frac {(a+b x)^{3/2} (A+B x)}{x^6} \, dx\) [408]

Optimal. Leaf size=172 \[ \frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a^2 x^2}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x}}{128 a^3 x}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}+\frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{7/2}} \]

[Out]

1/8*(A*b-2*B*a)*(b*x+a)^(3/2)/a/x^4-1/5*A*(b*x+a)^(5/2)/a/x^5+3/128*b^4*(A*b-2*B*a)*arctanh((b*x+a)^(1/2)/a^(1
/2))/a^(7/2)+1/16*b*(A*b-2*B*a)*(b*x+a)^(1/2)/x^3/a+1/64*b^2*(A*b-2*B*a)*(b*x+a)^(1/2)/a^2/x^2-3/128*b^3*(A*b-
2*B*a)*(b*x+a)^(1/2)/a^3/x

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Rubi [A]
time = 0.05, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 44, 65, 214} \begin {gather*} \frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{7/2}}-\frac {3 b^3 \sqrt {a+b x} (A b-2 a B)}{128 a^3 x}+\frac {b^2 \sqrt {a+b x} (A b-2 a B)}{64 a^2 x^2}+\frac {(a+b x)^{3/2} (A b-2 a B)}{8 a x^4}+\frac {b \sqrt {a+b x} (A b-2 a B)}{16 a x^3}-\frac {A (a+b x)^{5/2}}{5 a x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/x^6,x]

[Out]

(b*(A*b - 2*a*B)*Sqrt[a + b*x])/(16*a*x^3) + (b^2*(A*b - 2*a*B)*Sqrt[a + b*x])/(64*a^2*x^2) - (3*b^3*(A*b - 2*
a*B)*Sqrt[a + b*x])/(128*a^3*x) + ((A*b - 2*a*B)*(a + b*x)^(3/2))/(8*a*x^4) - (A*(a + b*x)^(5/2))/(5*a*x^5) +
(3*b^4*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(7/2))

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 79

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] || L
tQ[p, n]))))

Rule 214

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

Rubi steps

\begin {align*} \int \frac {(a+b x)^{3/2} (A+B x)}{x^6} \, dx &=-\frac {A (a+b x)^{5/2}}{5 a x^5}+\frac {\left (-\frac {5 A b}{2}+5 a B\right ) \int \frac {(a+b x)^{3/2}}{x^5} \, dx}{5 a}\\ &=\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}-\frac {(3 b (A b-2 a B)) \int \frac {\sqrt {a+b x}}{x^4} \, dx}{16 a}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}-\frac {\left (b^2 (A b-2 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{32 a}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a^2 x^2}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}+\frac {\left (3 b^3 (A b-2 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{128 a^2}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a^2 x^2}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x}}{128 a^3 x}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}-\frac {\left (3 b^4 (A b-2 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{256 a^3}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a^2 x^2}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x}}{128 a^3 x}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}-\frac {\left (3 b^3 (A b-2 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{128 a^3}\\ &=\frac {b (A b-2 a B) \sqrt {a+b x}}{16 a x^3}+\frac {b^2 (A b-2 a B) \sqrt {a+b x}}{64 a^2 x^2}-\frac {3 b^3 (A b-2 a B) \sqrt {a+b x}}{128 a^3 x}+\frac {(A b-2 a B) (a+b x)^{3/2}}{8 a x^4}-\frac {A (a+b x)^{5/2}}{5 a x^5}+\frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.28, size = 128, normalized size = 0.74 \begin {gather*} -\frac {\sqrt {a+b x} \left (15 A b^4 x^4-10 a b^3 x^3 (A+3 B x)+4 a^2 b^2 x^2 (2 A+5 B x)+32 a^4 (4 A+5 B x)+16 a^3 b x (11 A+15 B x)\right )}{640 a^3 x^5}+\frac {3 b^4 (A b-2 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/x^6,x]

[Out]

-1/640*(Sqrt[a + b*x]*(15*A*b^4*x^4 - 10*a*b^3*x^3*(A + 3*B*x) + 4*a^2*b^2*x^2*(2*A + 5*B*x) + 32*a^4*(4*A + 5
*B*x) + 16*a^3*b*x*(11*A + 15*B*x)))/(a^3*x^5) + (3*b^4*(A*b - 2*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(128*a^(
7/2))

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Maple [A]
time = 0.08, size = 130, normalized size = 0.76

method	result	size
derivativedivides	\(2 b^{4} \left (-\frac {\frac {3 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{3}}-\frac {7 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{2}}+\frac {A b \left (b x +a \right )^{\frac {5}{2}}}{10 a}+\left (\frac {7 A b}{128}-\frac {7 B a}{64}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {3 a \left (A b -2 B a \right ) \sqrt {b x +a}}{256}}{b^{5} x^{5}}+\frac {3 \left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {7}{2}}}\right )\)	\(130\)
default	\(2 b^{4} \left (-\frac {\frac {3 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{3}}-\frac {7 \left (A b -2 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{128 a^{2}}+\frac {A b \left (b x +a \right )^{\frac {5}{2}}}{10 a}+\left (\frac {7 A b}{128}-\frac {7 B a}{64}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {3 a \left (A b -2 B a \right ) \sqrt {b x +a}}{256}}{b^{5} x^{5}}+\frac {3 \left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {7}{2}}}\right )\)	\(130\)
risch	\(-\frac {\sqrt {b x +a}\, \left (15 A \,b^{4} x^{4}-30 B a \,b^{3} x^{4}-10 A a \,b^{3} x^{3}+20 B \,a^{2} b^{2} x^{3}+8 A \,a^{2} b^{2} x^{2}+240 B \,a^{3} b \,x^{2}+176 A \,a^{3} b x +160 B \,a^{4} x +128 A \,a^{4}\right )}{640 x^{5} a^{3}}+\frac {3 b^{4} \left (A b -2 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {7}{2}}}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/x^6,x,method=_RETURNVERBOSE)

[Out]

2*b^4*(-(3/256*(A*b-2*B*a)/a^3*(b*x+a)^(9/2)-7/128*(A*b-2*B*a)/a^2*(b*x+a)^(7/2)+1/10*A*b/a*(b*x+a)^(5/2)+(7/1
28*A*b-7/64*B*a)*(b*x+a)^(3/2)-3/256*a*(A*b-2*B*a)*(b*x+a)^(1/2))/b^5/x^5+3/256*(A*b-2*B*a)/a^(7/2)*arctanh((b
*x+a)^(1/2)/a^(1/2)))

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Maxima [A]
time = 0.49, size = 224, normalized size = 1.30 \begin {gather*} -\frac {1}{1280} \, b^{5} {\left (\frac {2 \, {\left (128 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b - 15 \, {\left (2 \, B a - A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 70 \, {\left (2 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 70 \, {\left (2 \, B a^{4} - A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (2 \, B a^{5} - A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{3} b - 5 \, {\left (b x + a\right )}^{4} a^{4} b + 10 \, {\left (b x + a\right )}^{3} a^{5} b - 10 \, {\left (b x + a\right )}^{2} a^{6} b + 5 \, {\left (b x + a\right )} a^{7} b - a^{8} b} - \frac {15 \, {\left (2 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {7}{2}} b}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^6,x, algorithm="maxima")

[Out]

-1/1280*b^5*(2*(128*(b*x + a)^(5/2)*A*a^2*b - 15*(2*B*a - A*b)*(b*x + a)^(9/2) + 70*(2*B*a^2 - A*a*b)*(b*x + a
)^(7/2) - 70*(2*B*a^4 - A*a^3*b)*(b*x + a)^(3/2) + 15*(2*B*a^5 - A*a^4*b)*sqrt(b*x + a))/((b*x + a)^5*a^3*b -
5*(b*x + a)^4*a^4*b + 10*(b*x + a)^3*a^5*b - 10*(b*x + a)^2*a^6*b + 5*(b*x + a)*a^7*b - a^8*b) - 15*(2*B*a - A
*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(7/2)*b))

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Fricas [A]
time = 1.14, size = 306, normalized size = 1.78 \begin {gather*} \left [-\frac {15 \, {\left (2 \, B a b^{4} - A b^{5}\right )} \sqrt {a} x^{5} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (128 \, A a^{5} - 15 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} x^{4} + 10 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2}\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1280 \, a^{4} x^{5}}, \frac {15 \, {\left (2 \, B a b^{4} - A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (128 \, A a^{5} - 15 \, {\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} x^{4} + 10 \, {\left (2 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (30 \, B a^{4} b + A a^{3} b^{2}\right )} x^{2} + 16 \, {\left (10 \, B a^{5} + 11 \, A a^{4} b\right )} x\right )} \sqrt {b x + a}}{640 \, a^{4} x^{5}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^6,x, algorithm="fricas")

[Out]

[-1/1280*(15*(2*B*a*b^4 - A*b^5)*sqrt(a)*x^5*log((b*x + 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*(128*A*a^5 - 15*
(2*B*a^2*b^3 - A*a*b^4)*x^4 + 10*(2*B*a^3*b^2 - A*a^2*b^3)*x^3 + 8*(30*B*a^4*b + A*a^3*b^2)*x^2 + 16*(10*B*a^5
+ 11*A*a^4*b)*x)*sqrt(b*x + a))/(a^4*x^5), 1/640*(15*(2*B*a*b^4 - A*b^5)*sqrt(-a)*x^5*arctan(sqrt(b*x + a)*sq
rt(-a)/a) - (128*A*a^5 - 15*(2*B*a^2*b^3 - A*a*b^4)*x^4 + 10*(2*B*a^3*b^2 - A*a^2*b^3)*x^3 + 8*(30*B*a^4*b + A
*a^3*b^2)*x^2 + 16*(10*B*a^5 + 11*A*a^4*b)*x)*sqrt(b*x + a))/(a^4*x^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1843 vs. \(2 (158) = 316\).
time = 84.08, size = 1843, normalized size = 10.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(B*x+A)/x**6,x)

[Out]

-1930*A*a**6*b**5*sqrt(a + b*x)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**
3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) + 4740*A*a**5*b**5*(a + b*x)**(3/2)/(5120*a**10 + 6400*a*
*9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5)
- 5376*A*a**4*b**5*(a + b*x)**(5/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*
x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) - 1116*A*a**4*b**5*sqrt(a + b*x)/(-1152*a**8 - 1536*a
**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 2940*A*a**3*b**5*(a + b*x
)**(7/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)
**4 + 1280*a**5*(a + b*x)**5) + 2044*A*a**3*b**5*(a + b*x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a +
b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 630*A*a**2*b**5*(a + b*x)**(9/2)/(5120*a**10 + 64
00*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)
**5) - 1540*A*a**2*b**5*(a + b*x)**(5/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a +
b*x)**3 + 384*a**4*(a + b*x)**4) - 66*A*a**2*b**5*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)*
*2 + 48*a**3*(a + b*x)**3) - 63*A*a**2*b**5*sqrt(a**(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 63*
A*a**2*b**5*sqrt(a**(-11))*log(a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 420*A*a*b**5*(a + b*x)**(7/2)/(-1152
*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 80*A*a*b**5
*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) + 35*A*a*b**5*sqrt(a
**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/64 - 35*A*a*b**5*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(
a + b*x))/64 - 30*A*b**5*(a + b*x)**(5/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)*
*3) - 5*A*b**5*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 + 5*A*b**5*sqrt(a**(-7))*log(a**4*sqr
t(a**(-7)) + sqrt(a + b*x))/16 - 558*B*a**5*b**4*sqrt(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*
x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 1022*B*a**4*b**4*(a + b*x)**(3/2)/(-1152*a**8 - 1536
*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) - 770*B*a**3*b**4*(a + b*
x)**(5/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**
4) - 132*B*a**3*b**4*sqrt(a + b*x)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) + 2
10*B*a**2*b**4*(a + b*x)**(7/2)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3
+ 384*a**4*(a + b*x)**4) + 160*B*a**2*b**4*(a + b*x)**(3/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*x)**2 +
48*a**3*(a + b*x)**3) + 35*B*a**2*b**4*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 35*B*a**2*
b**4*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 60*B*a*b**4*(a + b*x)**(5/2)/(96*a**6 + 144*a
**5*b*x - 144*a**4*(a + b*x)**2 + 48*a**3*(a + b*x)**3) - 10*B*a*b**4*sqrt(a + b*x)/(-8*a**4 - 16*a**3*b*x + 8
*a**2*(a + b*x)**2) - 5*B*a*b**4*sqrt(a**(-7))*log(-a**4*sqrt(a**(-7)) + sqrt(a + b*x))/8 + 5*B*a*b**4*sqrt(a*
*(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/8 + 6*B*b**4*(a + b*x)**(3/2)/(-8*a**4 - 16*a**3*b*x + 8*a**2*(
a + b*x)**2) + 3*B*b**4*sqrt(a**(-5))*log(-a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8 - 3*B*b**4*sqrt(a**(-5))*log(
a**3*sqrt(a**(-5)) + sqrt(a + b*x))/8

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Giac [A]
time = 2.65, size = 192, normalized size = 1.12 \begin {gather*} \frac {\frac {15 \, {\left (2 \, B a b^{5} - A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3}} + \frac {30 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 140 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 140 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 30 \, \sqrt {b x + a} B a^{5} b^{5} - 15 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 70 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 128 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} - 70 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 15 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{3} b^{5} x^{5}}}{640 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^6,x, algorithm="giac")

[Out]

1/640*(15*(2*B*a*b^5 - A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^3) + (30*(b*x + a)^(9/2)*B*a*b^5 - 14
0*(b*x + a)^(7/2)*B*a^2*b^5 + 140*(b*x + a)^(3/2)*B*a^4*b^5 - 30*sqrt(b*x + a)*B*a^5*b^5 - 15*(b*x + a)^(9/2)*
A*b^6 + 70*(b*x + a)^(7/2)*A*a*b^6 - 128*(b*x + a)^(5/2)*A*a^2*b^6 - 70*(b*x + a)^(3/2)*A*a^3*b^6 + 15*sqrt(b*
x + a)*A*a^4*b^6)/(a^3*b^5*x^5))/b

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Mupad [B]
time = 0.44, size = 203, normalized size = 1.18 \begin {gather*} \frac {\left (\frac {7\,A\,b^5}{64}-\frac {7\,B\,a\,b^4}{32}\right )\,{\left (a+b\,x\right )}^{3/2}+\left (\frac {3\,B\,a^2\,b^4}{64}-\frac {3\,A\,a\,b^5}{128}\right )\,\sqrt {a+b\,x}-\frac {7\,\left (A\,b^5-2\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{64\,a^2}+\frac {3\,\left (A\,b^5-2\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^3}+\frac {A\,b^5\,{\left (a+b\,x\right )}^{5/2}}{5\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}+\frac {3\,b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-2\,B\,a\right )}{128\,a^{7/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/x^6,x)

[Out]

(((7*A*b^5)/64 - (7*B*a*b^4)/32)*(a + b*x)^(3/2) + ((3*B*a^2*b^4)/64 - (3*A*a*b^5)/128)*(a + b*x)^(1/2) - (7*(
A*b^5 - 2*B*a*b^4)*(a + b*x)^(7/2))/(64*a^2) + (3*(A*b^5 - 2*B*a*b^4)*(a + b*x)^(9/2))/(128*a^3) + (A*b^5*(a +
b*x)^(5/2))/(5*a))/(5*a*(a + b*x)^4 - 5*a^4*(a + b*x) - (a + b*x)^5 - 10*a^2*(a + b*x)^3 + 10*a^3*(a + b*x)^2
+ a^5) + (3*b^4*atanh((a + b*x)^(1/2)/a^(1/2))*(A*b - 2*B*a))/(128*a^(7/2))

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