3.5.21 \(\int \frac {(a+b x)^{5/2} (A+B x)}{x^7} \, dx\) [421]

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

[Out]

1/96*b*(5*A*b-12*B*a)*(b*x+a)^(3/2)/a/x^4+1/60*(5*A*b-12*B*a)*(b*x+a)^(5/2)/a/x^5-1/6*A*(b*x+a)^(7/2)/a/x^6+1/
512*b^5*(5*A*b-12*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(7/2)+1/192*b^2*(5*A*b-12*B*a)*(b*x+a)^(1/2)/x^3/a+1/7
68*b^3*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a^2/x^2-1/512*b^4*(5*A*b-12*B*a)*(b*x+a)^(1/2)/a^3/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 (5 A b-12 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{7/2}}-\frac {b^4 \sqrt {a+b x} (5 A b-12 a B)}{512 a^3 x}+\frac {b^3 \sqrt {a+b x} (5 A b-12 a B)}{768 a^2 x^2}+\frac {b^2 \sqrt {a+b x} (5 A b-12 a B)}{192 a x^3}+\frac {(a+b x)^{5/2} (5 A b-12 a B)}{60 a x^5}+\frac {b (a+b x)^{3/2} (5 A b-12 a B)}{96 a x^4}-\frac {A (a+b x)^{7/2}}{6 a x^6} \end {gather*}

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 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)^{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 ) \text {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*}

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Mathematica [A]
time = 0.34, size = 148, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {a+b x} \left (75 A b^5 x^5+40 a^2 b^3 x^3 (A+3 B x)+256 a^5 (5 A+6 B x)-10 a b^4 x^4 (5 A+18 B x)+48 a^3 b^2 x^2 (45 A+62 B x)+64 a^4 b x (50 A+63 B x)\right )}{7680 a^3 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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7680*(Sqrt[a + b*x]*(75*A*b^5*x^5 + 40*a^2*b^3*x^3*(A + 3*B*x) + 256*a^5*(5*A + 6*B*x) - 10*a*b^4*x^4*(5*A
+ 18*B*x) + 48*a^3*b^2*x^2*(45*A + 62*B*x) + 64*a^4*b*x*(50*A + 63*B*x)))/(a^3*x^6) + (b^5*(5*A*b - 12*a*B)*Ar
cTanh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(7/2))

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Maple [A]
time = 0.07, size = 162, normalized size = 0.78

method	result	size
risch	\(-\frac {\sqrt {b x +a}\, \left (75 A \,b^{5} x^{5}-180 B a \,b^{4} x^{5}-50 a A \,b^{4} x^{4}+120 B \,a^{2} b^{3} x^{4}+40 a^{2} A \,b^{3} x^{3}+2976 B \,a^{3} b^{2} x^{3}+2160 a^{3} A \,b^{2} x^{2}+4032 B \,a^{4} b \,x^{2}+3200 a^{4} A b x +1536 a^{5} B x +1280 a^{5} A \right )}{7680 x^{6} a^{3}}+\frac {b^{5} \left (5 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{512 a^{\frac {7}{2}}}\)	\(155\)
derivativedivides	\(2 b^{5} \left (-\frac {\frac {\left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{3}}-\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{2}}+\frac {\left (165 A b +116 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a}+\left (\frac {33 A b}{512}-\frac {99 B a}{640}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {17 a \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3072}+\frac {a^{2} \left (5 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {\left (5 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {7}{2}}}\right )\)	\(162\)
default	\(2 b^{5} \left (-\frac {\frac {\left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{3}}-\frac {17 \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{2}}+\frac {\left (165 A b +116 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a}+\left (\frac {33 A b}{512}-\frac {99 B a}{640}\right ) \left (b x +a \right )^{\frac {5}{2}}-\frac {17 a \left (5 A b -12 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{3072}+\frac {a^{2} \left (5 A b -12 B a \right ) \sqrt {b x +a}}{1024}}{b^{6} x^{6}}+\frac {\left (5 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {7}{2}}}\right )\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)))

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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 - 5 \, A b\right )} {\left (b x + a\right )}^{\frac {11}{2}} - 85 \, {\left (12 \, B a^{2} - 5 \, A a b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 6 \, {\left (116 \, B a^{3} + 165 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 198 \, {\left (12 \, B a^{4} - 5 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 85 \, {\left (12 \, B a^{5} - 5 \, A a^{4} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 15 \, {\left (12 \, B a^{6} - 5 \, A a^{5} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{6} a^{3} b - 6 \, {\left (b x + a\right )}^{5} a^{4} b + 15 \, {\left (b x + a\right )}^{4} a^{5} b - 20 \, {\left (b x + a\right )}^{3} a^{6} b + 15 \, {\left (b x + a\right )}^{2} a^{7} b - 6 \, {\left (b x + a\right )} a^{8} b + a^{9} b} + \frac {15 \, {\left (12 \, B a - 5 \, 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)^(5/2)*(B*x+A)/x^7,x, algorithm="maxima")

[Out]

1/15360*b^6*(2*(15*(12*B*a - 5*A*b)*(b*x + a)^(11/2) - 85*(12*B*a^2 - 5*A*a*b)*(b*x + a)^(9/2) - 6*(116*B*a^3
+ 165*A*a^2*b)*(b*x + a)^(7/2) + 198*(12*B*a^4 - 5*A*a^3*b)*(b*x + a)^(5/2) - 85*(12*B*a^5 - 5*A*a^4*b)*(b*x +
a)^(3/2) + 15*(12*B*a^6 - 5*A*a^5*b)*sqrt(b*x + a))/((b*x + a)^6*a^3*b - 6*(b*x + a)^5*a^4*b + 15*(b*x + a)^4
*a^5*b - 20*(b*x + a)^3*a^6*b + 15*(b*x + a)^2*a^7*b - 6*(b*x + a)*a^8*b + a^9*b) + 15*(12*B*a - 5*A*b)*log((s
qrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(7/2)*b))

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Fricas [A]
time = 0.87, size = 356, normalized size = 1.71 \begin {gather*} \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 {gather*}

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)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2963 vs. \(2 (190) = 380\).
time = 147.51, size = 2963, normalized size = 14.25 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23790*A*a**8*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**7*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**6*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) - 5790*A*a**6*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**5*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) + 14220*A*a**
5*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**4*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) - 16128*A*a**4*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) - 1674*A*a**4*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**3*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) + 8820*A*a**3
*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) + 3066*A*a**3*b**6*(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) + 231*A*a**3*b**6*sqrt(a**(-13))*log(
-a**7*sqrt(a**(-13)) + sqrt(a + b*x))/1024 - 231*A*a**3*b**6*sqrt(a**(-13))*log(a**7*sqrt(a**(-13)) + sqrt(a +
b*x))/1024 - 1890*A*a**2*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) - 2310*A*a**2*b**6*(a + b*x)**(5/2)/(-115
2*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**6*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) - 189*A*a**2*b**6*s
qrt(a**(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 189*A*a**2*b**6*sqrt(a**(-11))*log(a**6*sqrt(a**
(-11)) + sqrt(a + b*x))/256 + 630*A*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) + 80*A*a*b**6*(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) + 105*A*a*b**6*sqrt(a**(-9))*log(-a**5*sqrt(a**(-9)) + sqrt(a +
b*x))/128 - 105*A*a*b**6*sqrt(a**(-9))*log(a**5*sqrt(a**(-9)) + sqrt(a + b*x))/128 - 30*A*b**6*(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**6*sqrt(a**(-7))*log(-a**4*s
qrt(a**(-7)) + sqrt(a + b*x))/16 + 5*A*b**6*sqrt(a**(-7))*log(a**4*sqrt(a**(-7)) + sqrt(a + b*x))/16 - 1930*B*
a**7*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**6*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*B
*a**5*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) - 1674*B*a**5*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*B*a**4*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 + 12
80*a**5*(a + b*x)**5) + 3066*B*a**4*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*B*a**3*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) - 2
310*B*a**3*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) - 198*B*a**3*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**3*b**5*sqrt(a**(-11))*log(-a**6*sqrt(a**(-11)) + sqrt(a + b*x))/256 + 63*B*a**3*
b**5*sqrt(a**(-11))*log(a**6*sqrt(a**(-11)) + s...

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Giac [A]
time = 1.06, size = 240, normalized size = 1.15 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.47, size = 254, normalized size = 1.22 \begin {gather*} \frac {b^5\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (5\,A\,b-12\,B\,a\right )}{512\,a^{7/2}}-\frac {\left (\frac {33\,A\,b^6}{256}-\frac {99\,B\,a\,b^5}{320}\right )\,{\left (a+b\,x\right )}^{5/2}+\left (\frac {5\,A\,a^2\,b^6}{512}-\frac {3\,B\,a^3\,b^5}{128}\right )\,\sqrt {a+b\,x}+\left (\frac {17\,B\,a^2\,b^5}{128}-\frac {85\,A\,a\,b^6}{1536}\right )\,{\left (a+b\,x\right )}^{3/2}-\frac {17\,\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{9/2}}{1536\,a^2}+\frac {\left (5\,A\,b^6-12\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{11/2}}{512\,a^3}+\frac {\left (165\,A\,b^6+116\,B\,a\,b^5\right )\,{\left (a+b\,x\right )}^{7/2}}{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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*atanh((a + b*x)^(1/2)/a^(1/2))*(5*A*b - 12*B*a))/(512*a^(7/2)) - (((33*A*b^6)/256 - (99*B*a*b^5)/320)*(a
+ b*x)^(5/2) + ((5*A*a^2*b^6)/512 - (3*B*a^3*b^5)/128)*(a + b*x)^(1/2) + ((17*B*a^2*b^5)/128 - (85*A*a*b^6)/15
36)*(a + b*x)^(3/2) - (17*(5*A*b^6 - 12*B*a*b^5)*(a + b*x)^(9/2))/(1536*a^2) + ((5*A*b^6 - 12*B*a*b^5)*(a + b*
x)^(11/2))/(512*a^3) + ((165*A*b^6 + 116*B*a*b^5)*(a + b*x)^(7/2))/(1280*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)

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