3.5.9 \(\int \frac {(a+b x)^{3/2} (A+B x)}{x^7} \, dx\) [409]

Optimal. Leaf size=208 \[ \frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}} \]

[Out]

1/60*(7*A*b-12*B*a)*(b*x+a)^(3/2)/a/x^5-1/6*A*(b*x+a)^(5/2)/a/x^6-1/512*b^5*(7*A*b-12*B*a)*arctanh((b*x+a)^(1/
2)/a^(1/2))/a^(9/2)+1/160*b*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a/x^4+1/960*b^2*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^2/x^3-
1/768*b^3*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^3/x^2+1/512*b^4*(7*A*b-12*B*a)*(b*x+a)^(1/2)/a^4/x

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Rubi [A]
time = 0.07, antiderivative size = 208, normalized size of antiderivative = 1.00, 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 = {79, 43, 44, 65, 214} \begin {gather*} -\frac {b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}}+\frac {b^4 \sqrt {a+b x} (7 A b-12 a B)}{512 a^4 x}-\frac {b^3 \sqrt {a+b x} (7 A b-12 a B)}{768 a^3 x^2}+\frac {b^2 \sqrt {a+b x} (7 A b-12 a B)}{960 a^2 x^3}+\frac {(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}+\frac {b \sqrt {a+b x} (7 A b-12 a B)}{160 a x^4}-\frac {A (a+b x)^{5/2}}{6 a x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x))/x^7,x]

[Out]

(b*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(160*a*x^4) + (b^2*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(960*a^2*x^3) - (b^3*(7*
A*b - 12*a*B)*Sqrt[a + b*x])/(768*a^3*x^2) + (b^4*(7*A*b - 12*a*B)*Sqrt[a + b*x])/(512*a^4*x) + ((7*A*b - 12*a
*B)*(a + b*x)^(3/2))/(60*a*x^5) - (A*(a + b*x)^(5/2))/(6*a*x^6) - (b^5*(7*A*b - 12*a*B)*ArcTanh[Sqrt[a + b*x]/
Sqrt[a]])/(512*a^(9/2))

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^7} \, dx &=-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (-\frac {7 A b}{2}+6 a B\right ) \int \frac {(a+b x)^{3/2}}{x^6} \, dx}{6 a}\\ &=\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {(b (7 A b-12 a B)) \int \frac {\sqrt {a+b x}}{x^5} \, dx}{40 a}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {\left (b^2 (7 A b-12 a B)\right ) \int \frac {1}{x^4 \sqrt {a+b x}} \, dx}{320 a}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^3 (7 A b-12 a B)\right ) \int \frac {1}{x^3 \sqrt {a+b x}} \, dx}{384 a^2}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {\left (b^4 (7 A b-12 a B)\right ) \int \frac {1}{x^2 \sqrt {a+b x}} \, dx}{512 a^3}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^5 (7 A b-12 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{1024 a^4}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}+\frac {\left (b^4 (7 A b-12 a B)\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{512 a^4}\\ &=\frac {b (7 A b-12 a B) \sqrt {a+b x}}{160 a x^4}+\frac {b^2 (7 A b-12 a B) \sqrt {a+b x}}{960 a^2 x^3}-\frac {b^3 (7 A b-12 a B) \sqrt {a+b x}}{768 a^3 x^2}+\frac {b^4 (7 A b-12 a B) \sqrt {a+b x}}{512 a^4 x}+\frac {(7 A b-12 a B) (a+b x)^{3/2}}{60 a x^5}-\frac {A (a+b x)^{5/2}}{6 a x^6}-\frac {b^5 (7 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.33, size = 148, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {a+b x} \left (-105 A b^5 x^5+48 a^3 b^2 x^2 (A+2 B x)+256 a^5 (5 A+6 B x)-8 a^2 b^3 x^3 (7 A+15 B x)+10 a b^4 x^4 (7 A+18 B x)+64 a^4 b x (26 A+33 B x)\right )}{7680 a^4 x^6}+\frac {b^5 (-7 A b+12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x))/x^7,x]

[Out]

-1/7680*(Sqrt[a + b*x]*(-105*A*b^5*x^5 + 48*a^3*b^2*x^2*(A + 2*B*x) + 256*a^5*(5*A + 6*B*x) - 8*a^2*b^3*x^3*(7
*A + 15*B*x) + 10*a*b^4*x^4*(7*A + 18*B*x) + 64*a^4*b*x*(26*A + 33*B*x)))/(a^4*x^6) + (b^5*(-7*A*b + 12*a*B)*A
rcTanh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(9/2))

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Maple [A]
time = 0.08, size = 162, normalized size = 0.78

method	result	size
risch	\(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{5} x^{5}+180 B a \,b^{4} x^{5}+70 a A \,b^{4} x^{4}-120 B \,a^{2} b^{3} x^{4}-56 a^{2} A \,b^{3} x^{3}+96 B \,a^{3} b^{2} x^{3}+48 a^{3} A \,b^{2} x^{2}+2112 B \,a^{4} b \,x^{2}+1664 a^{4} A b x +1536 a^{5} B x +1280 a^{5} A \right )}{7680 x^{6} a^{4}}-\frac {b^{5} \left (7 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {9}{2}}}\)	\(155\)
derivativedivides	\(2 b^{5} \left (-\frac {-\frac {\left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{4}}+\frac {17 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{3}}-\frac {33 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a^{2}}+\frac {\left (281 A b -116 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{2560 a}+\left (\frac {119 A b}{3072}-\frac {17 B a}{256}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {a \left (7 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}-\frac {\left (7 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}\right )\)	\(162\)
default	\(2 b^{5} \left (-\frac {-\frac {\left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{4}}+\frac {17 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{3}}-\frac {33 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a^{2}}+\frac {\left (281 A b -116 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{2560 a}+\left (\frac {119 A b}{3072}-\frac {17 B a}{256}\right ) \left (b x +a \right )^{\frac {3}{2}}-\frac {a \left (7 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}-\frac {\left (7 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}\right )\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(B*x+A)/x^7,x,method=_RETURNVERBOSE)

[Out]

2*b^5*(-(-1/1024*(7*A*b-12*B*a)/a^4*(b*x+a)^(11/2)+17/3072/a^3*(7*A*b-12*B*a)*(b*x+a)^(9/2)-33/2560/a^2*(7*A*b
-12*B*a)*(b*x+a)^(7/2)+1/2560*(281*A*b-116*B*a)/a*(b*x+a)^(5/2)+(119/3072*A*b-17/256*B*a)*(b*x+a)^(3/2)-1/1024
*a*(7*A*b-12*B*a)*(b*x+a)^(1/2))/b^6/x^6-1/1024*(7*A*b-12*B*a)/a^(9/2)*arctanh((b*x+a)^(1/2)/a^(1/2)))

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Maxima [A]
time = 0.50, size = 268, normalized size = 1.29 \begin {gather*} -\frac {1}{15360} \, b^{6} {\left (\frac {2 \, {\left (15 \, {\left (12 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 85 \, {\left (12 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} + 198 \, {\left (12 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} - 6 \, {\left (116 \, B a^{4} - 281 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 85 \, {\left (12 \, B a^{5} - 7 \, A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (12 \, B a^{6} - 7 \, A a^{5} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{6} a^{4} b - 6 \, {\left (b x + a\right )}^{5} a^{5} b + 15 \, {\left (b x + a\right )}^{4} a^{6} b - 20 \, {\left (b x + a\right )}^{3} a^{7} b + 15 \, {\left (b x + a\right )}^{2} a^{8} b - 6 \, {\left (b x + a\right )} a^{9} b + a^{10} b} + \frac {15 \, {\left (12 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{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^7,x, algorithm="maxima")

[Out]

-1/15360*b^6*(2*(15*(12*B*a - 7*A*b)*(b*x + a)^(11/2) - 85*(12*B*a^2 - 7*A*a*b)*(b*x + a)^(9/2) + 198*(12*B*a^
3 - 7*A*a^2*b)*(b*x + a)^(7/2) - 6*(116*B*a^4 - 281*A*a^3*b)*(b*x + a)^(5/2) - 85*(12*B*a^5 - 7*A*a^4*b)*(b*x
+ a)^(3/2) + 15*(12*B*a^6 - 7*A*a^5*b)*sqrt(b*x + a))/((b*x + a)^6*a^4*b - 6*(b*x + a)^5*a^5*b + 15*(b*x + a)^
4*a^6*b - 20*(b*x + a)^3*a^7*b + 15*(b*x + a)^2*a^8*b - 6*(b*x + a)*a^9*b + a^10*b) + 15*(12*B*a - 7*A*b)*log(
(sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(9/2)*b))

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Fricas [A]
time = 1.05, size = 353, normalized size = 1.70 \begin {gather*} \left [-\frac {15 \, {\left (12 \, B a b^{5} - 7 \, 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} - 7 \, A a b^{5}\right )} x^{5} - 10 \, {\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \, {\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{15360 \, a^{5} x^{6}}, -\frac {15 \, {\left (12 \, B a b^{5} - 7 \, 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} - 7 \, A a b^{5}\right )} x^{5} - 10 \, {\left (12 \, B a^{3} b^{3} - 7 \, A a^{2} b^{4}\right )} x^{4} + 8 \, {\left (12 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3}\right )} x^{3} + 48 \, {\left (44 \, B a^{5} b + A a^{4} b^{2}\right )} x^{2} + 128 \, {\left (12 \, B a^{6} + 13 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{7680 \, a^{5} x^{6}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^7,x, algorithm="fricas")

[Out]

[-1/15360*(15*(12*B*a*b^5 - 7*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 - 7*A*a*b^5)*x^5 - 10*(12*B*a^3*b^3 - 7*A*a^2*b^4)*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3)*x^3
+ 48*(44*B*a^5*b + A*a^4*b^2)*x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(b*x + a))/(a^5*x^6), -1/7680*(15*(12*
B*a*b^5 - 7*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 - 7*A*a*b^5)
*x^5 - 10*(12*B*a^3*b^3 - 7*A*a^2*b^4)*x^4 + 8*(12*B*a^4*b^2 - 7*A*a^3*b^3)*x^3 + 48*(44*B*a^5*b + A*a^4*b^2)*
x^2 + 128*(12*B*a^6 + 13*A*a^5*b)*x)*sqrt(b*x + a))/(a^5*x^6)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2536 vs. \(2 (194) = 388\).
time = 116.89, size = 2536, normalized size = 12.19 \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**7,x)

[Out]

-23790*A*a**7*b**6*sqrt(a + b*x)/(-76800*a**12 - 92160*a**11*b*x + 230400*a**10*(a + b*x)**2 - 307200*a**9*(a
+ b*x)**3 + 230400*a**8*(a + b*x)**4 - 92160*a**7*(a + b*x)**5 + 15360*a**6*(a + b*x)**6) + 73370*A*a**6*b**6*
(a + b*x)**(3/2)/(-76800*a**12 - 92160*a**11*b*x + 230400*a**10*(a + b*x)**2 - 307200*a**9*(a + b*x)**3 + 2304
00*a**8*(a + b*x)**4 - 92160*a**7*(a + b*x)**5 + 15360*a**6*(a + b*x)**6) - 111276*A*a**5*b**6*(a + b*x)**(5/2
)/(-76800*a**12 - 92160*a**11*b*x + 230400*a**10*(a + b*x)**2 - 307200*a**9*(a + b*x)**3 + 230400*a**8*(a + b*
x)**4 - 92160*a**7*(a + b*x)**5 + 15360*a**6*(a + b*x)**6) - 3860*A*a**5*b**6*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) + 91476*A*a**4*b**6*(a + b*x)**(7/2)/(-76800*a**12 - 92160*a**11*b*x + 230400*a**10*(a + b*x)**2 - 307200*a
**9*(a + b*x)**3 + 230400*a**8*(a + b*x)**4 - 92160*a**7*(a + b*x)**5 + 15360*a**6*(a + b*x)**6) + 9480*A*a**4
*b**6*(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) - 39270*A*a**3*b**6*(a + b*x)**(9/2)/(-76800*a**12 - 92160*a**11*b
*x + 230400*a**10*(a + b*x)**2 - 307200*a**9*(a + b*x)**3 + 230400*a**8*(a + b*x)**4 - 92160*a**7*(a + b*x)**5
+ 15360*a**6*(a + b*x)**6) - 10752*A*a**3*b**6*(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) - 558*A*a**3*b**6*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)
+ 6930*A*a**2*b**6*(a + b*x)**(11/2)/(-76800*a**12 - 92160*a**11*b*x + 230400*a**10*(a + b*x)**2 - 307200*a**
9*(a + b*x)**3 + 230400*a**8*(a + b*x)**4 - 92160*a**7*(a + b*x)**5 + 15360*a**6*(a + b*x)**6) + 5880*A*a**2*b
**6*(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) + 1022*A*a**2*b**6*(a + b*x)**(3/2)/(-1152*a**8 - 1536*a**7*b*x + 23
04*a**6*(a + b*x)**2 - 1536*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 231*A*a**2*b**6*sqrt(a**(-13))*log(-a
**7*sqrt(a**(-13)) + sqrt(a + b*x))/1024 - 231*A*a**2*b**6*sqrt(a**(-13))*log(a**7*sqrt(a**(-13)) + sqrt(a + b
*x))/1024 - 1260*A*a*b**6*(a + b*x)**(9/2)/(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) - 770*A*a*b**6*(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) - 63*A*a*b**6*sqrt(a*
*(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/128 + 63*A*a*b**6*sqrt(a**(-11))*log(a**6*sqrt(a**(-11)) + s
qrt(a + b*x))/128 + 210*A*b**6*(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) + 35*A*b**6*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a + b*x))/12
8 - 35*A*b**6*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 1930*B*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*B*a**5*b**5*(a + b*x)**(3/2)/(5120*a**10 + 6400*a**9*b*x - 12800*a**8*(a + b*x)**2 + 12
800*a**7*(a + b*x)**3 - 6400*a**6*(a + b*x)**4 + 1280*a**5*(a + b*x)**5) - 5376*B*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*B*a**4*b**5*sqrt(a + b*x)/(-1152*a**8 - 1536*a**7*b*x + 2304*a**6*(a + b*x)**2 - 15
36*a**5*(a + b*x)**3 + 384*a**4*(a + b*x)**4) + 2940*B*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*
B*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 + 3
84*a**4*(a + b*x)**4) - 630*B*a**2*b**5*(a + b*x)**(9/2)/(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) - 1540*B*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) - 6
6*B*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*B*a**
2*b**5*sqrt(a**(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 63*B*a**2*b**5*sqrt(a**(-11))*log(a**6*s
qrt(a**(-11)) + sqrt(a + b*x))/256 + 420*B*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*B*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*B*a*b**5*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqr
t(a + b*x))/64 - 35*B*a*b**5*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/64 - 30*B*b**5*(a + b*x)**(
5/2)/(96*a**6 + 144*a**5*b*x - 144*a**4*(a + b*...

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Giac [A]
time = 1.46, size = 240, normalized size = 1.15 \begin {gather*} -\frac {\frac {15 \, {\left (12 \, B a b^{6} - 7 \, A b^{7}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \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} + 2376 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{3} b^{6} - 696 \, {\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} - 105 \, {\left (b x + a\right )}^{\frac {11}{2}} A b^{7} + 595 \, {\left (b x + a\right )}^{\frac {9}{2}} A a b^{7} - 1386 \, {\left (b x + a\right )}^{\frac {7}{2}} A a^{2} b^{7} + 1686 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{3} b^{7} + 595 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{4} b^{7} - 105 \, \sqrt {b x + a} A a^{5} b^{7}}{a^{4} b^{6} x^{6}}}{7680 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(B*x+A)/x^7,x, algorithm="giac")

[Out]

-1/7680*(15*(12*B*a*b^6 - 7*A*b^7)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^4) + (180*(b*x + a)^(11/2)*B*a*b
^6 - 1020*(b*x + a)^(9/2)*B*a^2*b^6 + 2376*(b*x + a)^(7/2)*B*a^3*b^6 - 696*(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 - 105*(b*x + a)^(11/2)*A*b^7 + 595*(b*x + a)^(9/2)*A*a*b
^7 - 1386*(b*x + a)^(7/2)*A*a^2*b^7 + 1686*(b*x + a)^(5/2)*A*a^3*b^7 + 595*(b*x + a)^(3/2)*A*a^4*b^7 - 105*sqr
t(b*x + a)*A*a^5*b^7)/(a^4*b^6*x^6))/b

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Mupad [B]
time = 0.16, size = 253, normalized size = 1.22 \begin {gather*} -\frac {\left (\frac {119\,A\,b^6}{1536}-\frac {17\,B\,a\,b^5}{128}\right )\,{\left (a+b\,x\right )}^{3/2}+\left (\frac {3\,B\,a^2\,b^5}{128}-\frac {7\,A\,a\,b^6}{512}\right )\,\sqrt {a+b\,x}-\frac {33\,\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{7/2}}{1280\,a^2}+\frac {17\,\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{9/2}}{1536\,a^3}-\frac {\left (7\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{11/2}}{512\,a^4}+\frac {\left (281\,A\,b^6-116\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{5/2}}{1280\,a}}{{\left (a+b\,x\right )}^6-6\,a^5\,\left (a+b\,x\right )-6\,a\,{\left (a+b\,x\right )}^5+15\,a^2\,{\left (a+b\,x\right )}^4-20\,a^3\,{\left (a+b\,x\right )}^3+15\,a^4\,{\left (a+b\,x\right )}^2+a^6}-\frac {b^5\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-12\,B\,a\right )}{512\,a^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^(3/2))/x^7,x)

[Out]

- (((119*A*b^6)/1536 - (17*B*a*b^5)/128)*(a + b*x)^(3/2) + ((3*B*a^2*b^5)/128 - (7*A*a*b^6)/512)*(a + b*x)^(1/
2) - (33*(7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(7/2))/(1280*a^2) + (17*(7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(9/2))/(153
6*a^3) - ((7*A*b^6 - 12*B*a*b^5)*(a + b*x)^(11/2))/(512*a^4) + ((281*A*b^6 - 116*B*a*b^5)*(a + b*x)^(5/2))/(12
80*a))/((a + b*x)^6 - 6*a^5*(a + b*x) - 6*a*(a + b*x)^5 + 15*a^2*(a + b*x)^4 - 20*a^3*(a + b*x)^3 + 15*a^4*(a
+ b*x)^2 + a^6) - (b^5*atanh((a + b*x)^(1/2)/a^(1/2))*(7*A*b - 12*B*a))/(512*a^(9/2))

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