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

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

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Rubi [A]  time = 0.10, 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 = {78, 47, 51, 63, 208} \begin {gather*} -\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 {b^4 \sqrt {a+b x} (7 A b-12 a B)}{512 a^4 x}-\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 \sqrt {a+b x} (7 A b-12 a B)}{160 a x^4}+\frac {(a+b x)^{3/2} (7 A b-12 a B)}{60 a x^5}-\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 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 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 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)^{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 ) \operatorname {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 [C]  time = 0.03, size = 58, normalized size = 0.28 \begin {gather*} -\frac {(a+b x)^{5/2} \left (5 a^6 A+b^5 x^6 (7 A b-12 a B) \, _2F_1\left (\frac {5}{2},6;\frac {7}{2};\frac {b x}{a}+1\right )\right )}{30 a^7 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/30*((a + b*x)^(5/2)*(5*a^6*A + b^5*(7*A*b - 12*a*B)*x^6*Hypergeometric2F1[5/2, 6, 7/2, 1 + (b*x)/a]))/(a^7*
x^6)

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IntegrateAlgebraic [A]  time = 0.32, size = 200, normalized size = 0.96 \begin {gather*} \frac {\left (12 a b^5 B-7 A b^6\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{512 a^{9/2}}-\frac {\sqrt {a+b x} \left (180 a^6 B-105 a^5 A b-1020 a^5 B (a+b x)+595 a^4 A b (a+b x)-696 a^4 B (a+b x)^2+1686 a^3 A b (a+b x)^2+2376 a^3 B (a+b x)^3-1386 a^2 A b (a+b x)^3-1020 a^2 B (a+b x)^4+595 a A b (a+b x)^4-105 A b (a+b x)^5+180 a B (a+b x)^5\right )}{7680 a^4 b x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/7680*(Sqrt[a + b*x]*(-105*a^5*A*b + 180*a^6*B + 595*a^4*A*b*(a + b*x) - 1020*a^5*B*(a + b*x) + 1686*a^3*A*b
*(a + b*x)^2 - 696*a^4*B*(a + b*x)^2 - 1386*a^2*A*b*(a + b*x)^3 + 2376*a^3*B*(a + b*x)^3 + 595*a*A*b*(a + b*x)
^4 - 1020*a^2*B*(a + b*x)^4 - 105*A*b*(a + b*x)^5 + 180*a*B*(a + b*x)^5))/(a^4*b*x^6) + ((-7*A*b^6 + 12*a*b^5*
B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(512*a^(9/2))

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fricas [A]  time = 1.35, 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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giac [A]  time = 1.29, 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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maple [A]  time = 0.02, size = 161, normalized size = 0.77 \begin {gather*} 2 \left (-\frac {\left (7 A b -12 B a \right ) \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{1024 a^{\frac {9}{2}}}+\frac {\frac {\left (7 A b -12 B a \right ) \sqrt {b x +a}\, a}{1024}-\frac {\left (281 A b -116 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{2560 a}+\frac {33 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{2560 a^{2}}-\frac {17 \left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{3072 a^{3}}+\frac {\left (7 A b -12 B a \right ) \left (b x +a \right )^{\frac {11}{2}}}{1024 a^{4}}+\left (-\frac {119 A b}{3072}+\frac {17 B a}{256}\right ) \left (b x +a \right )^{\frac {3}{2}}}{b^{6} x^{6}}\right ) b^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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-1
2*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))/x^6/b^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 = 2.05, 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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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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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