3.8.57 \(\int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx\) [757]

Optimal. Leaf size=154 \[ \frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \]

[Out]

-1/3*a*(7*A*b-9*B*a)*x^(3/2)/b^4+1/5*(7*A*b-9*B*a)*x^(5/2)/b^3-1/7*(7*A*b-9*B*a)*x^(7/2)/a/b^2+(A*b-B*a)*x^(9/
2)/a/b/(b*x+a)-a^(5/2)*(7*A*b-9*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/b^(11/2)+a^2*(7*A*b-9*B*a)*x^(1/2)/b^5

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Rubi [A]
time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {27, 79, 52, 65, 211} \begin {gather*} -\frac {a^{5/2} (7 A b-9 a B) \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}+\frac {a^2 \sqrt {x} (7 A b-9 a B)}{b^5}-\frac {a x^{3/2} (7 A b-9 a B)}{3 b^4}+\frac {x^{5/2} (7 A b-9 a B)}{5 b^3}-\frac {x^{7/2} (7 A b-9 a B)}{7 a b^2}+\frac {x^{9/2} (A b-a B)}{a b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(7*A*b - 9*a*B)*Sqrt[x])/b^5 - (a*(7*A*b - 9*a*B)*x^(3/2))/(3*b^4) + ((7*A*b - 9*a*B)*x^(5/2))/(5*b^3) -
((7*A*b - 9*a*B)*x^(7/2))/(7*a*b^2) + ((A*b - a*B)*x^(9/2))/(a*b*(a + b*x)) - (a^(5/2)*(7*A*b - 9*a*B)*ArcTan[
(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(11/2)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 211

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

Rubi steps

\begin {align*} \int \frac {x^{7/2} (A+B x)}{a^2+2 a b x+b^2 x^2} \, dx &=\int \frac {x^{7/2} (A+B x)}{(a+b x)^2} \, dx\\ &=\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (\frac {7 A b}{2}-\frac {9 a B}{2}\right ) \int \frac {x^{7/2}}{a+b x} \, dx}{a b}\\ &=-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {(7 A b-9 a B) \int \frac {x^{5/2}}{a+b x} \, dx}{2 b^2}\\ &=\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {(a (7 A b-9 a B)) \int \frac {x^{3/2}}{a+b x} \, dx}{2 b^3}\\ &=-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}+\frac {\left (a^2 (7 A b-9 a B)\right ) \int \frac {\sqrt {x}}{a+b x} \, dx}{2 b^4}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \int \frac {1}{\sqrt {x} (a+b x)} \, dx}{2 b^5}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {\left (a^3 (7 A b-9 a B)\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {x}\right )}{b^5}\\ &=\frac {a^2 (7 A b-9 a B) \sqrt {x}}{b^5}-\frac {a (7 A b-9 a B) x^{3/2}}{3 b^4}+\frac {(7 A b-9 a B) x^{5/2}}{5 b^3}-\frac {(7 A b-9 a B) x^{7/2}}{7 a b^2}+\frac {(A b-a B) x^{9/2}}{a b (a+b x)}-\frac {a^{5/2} (7 A b-9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 128, normalized size = 0.83 \begin {gather*} \frac {\sqrt {x} \left (-945 a^4 B+105 a^3 b (7 A-6 B x)+6 b^4 x^3 (7 A+5 B x)+14 a^2 b^2 x (35 A+9 B x)-2 a b^3 x^2 (49 A+27 B x)\right )}{105 b^5 (a+b x)}+\frac {a^{5/2} (-7 A b+9 a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(-945*a^4*B + 105*a^3*b*(7*A - 6*B*x) + 6*b^4*x^3*(7*A + 5*B*x) + 14*a^2*b^2*x*(35*A + 9*B*x) - 2*a*b
^3*x^2*(49*A + 27*B*x)))/(105*b^5*(a + b*x)) + (a^(5/2)*(-7*A*b + 9*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^
(11/2)

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Maple [A]
time = 0.65, size = 130, normalized size = 0.84

method	result	size
derivativedivides	\(\frac {\frac {2 B \,x^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\)	\(130\)
default	\(\frac {\frac {2 B \,x^{\frac {7}{2}} b^{3}}{7}+\frac {2 A \,b^{3} x^{\frac {5}{2}}}{5}-\frac {4 B a \,b^{2} x^{\frac {5}{2}}}{5}-\frac {4 A a \,b^{2} x^{\frac {3}{2}}}{3}+2 B \,a^{2} b \,x^{\frac {3}{2}}+6 A \,a^{2} b \sqrt {x}-8 a^{3} B \sqrt {x}}{b^{5}}-\frac {2 a^{3} \left (\frac {\left (-\frac {A b}{2}+\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -9 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{5}}\)	\(130\)
risch	\(\frac {2 \left (15 b^{3} B \,x^{3}+21 A \,b^{3} x^{2}-42 B a \,b^{2} x^{2}-70 A a \,b^{2} x +105 B \,a^{2} b x +315 A \,a^{2} b -420 B \,a^{3}\right ) \sqrt {x}}{105 b^{5}}+\frac {a^{3} \sqrt {x}\, A}{b^{4} \left (b x +a \right )}-\frac {a^{4} \sqrt {x}\, B}{b^{5} \left (b x +a \right )}-\frac {7 a^{3} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) A}{b^{4} \sqrt {a b}}+\frac {9 a^{4} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right ) B}{b^{5} \sqrt {a b}}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^5*(1/7*B*x^(7/2)*b^3+1/5*A*b^3*x^(5/2)-2/5*B*a*b^2*x^(5/2)-2/3*A*a*b^2*x^(3/2)+B*a^2*b*x^(3/2)+3*A*a^2*b*x
^(1/2)-4*a^3*B*x^(1/2))-2/b^5*a^3*((-1/2*A*b+1/2*B*a)*x^(1/2)/(b*x+a)+1/2*(7*A*b-9*B*a)/(a*b)^(1/2)*arctan(b*x
^(1/2)/(a*b)^(1/2)))

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Maxima [A]
time = 0.51, size = 139, normalized size = 0.90 \begin {gather*} -\frac {{\left (B a^{4} - A a^{3} b\right )} \sqrt {x}}{b^{6} x + a b^{5}} + \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} + \frac {2 \, {\left (15 \, B b^{3} x^{\frac {7}{2}} - 21 \, {\left (2 \, B a b^{2} - A b^{3}\right )} x^{\frac {5}{2}} + 35 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{\frac {3}{2}} - 105 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} \sqrt {x}\right )}}{105 \, b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B*a^4 - A*a^3*b)*sqrt(x)/(b^6*x + a*b^5) + (9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5)
+ 2/105*(15*B*b^3*x^(7/2) - 21*(2*B*a*b^2 - A*b^3)*x^(5/2) + 35*(3*B*a^2*b - 2*A*a*b^2)*x^(3/2) - 105*(4*B*a^
3 - 3*A*a^2*b)*sqrt(x))/b^5

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Fricas [A]
time = 2.71, size = 341, normalized size = 2.21 \begin {gather*} \left [-\frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x - 2 \, b \sqrt {x} \sqrt {-\frac {a}{b}} - a}{b x + a}\right ) - 2 \, {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{210 \, {\left (b^{6} x + a b^{5}\right )}}, \frac {105 \, {\left (9 \, B a^{4} - 7 \, A a^{3} b + {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b \sqrt {x} \sqrt {\frac {a}{b}}}{a}\right ) + {\left (30 \, B b^{4} x^{4} - 945 \, B a^{4} + 735 \, A a^{3} b - 6 \, {\left (9 \, B a b^{3} - 7 \, A b^{4}\right )} x^{3} + 14 \, {\left (9 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{2} - 70 \, {\left (9 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x\right )} \sqrt {x}}{105 \, {\left (b^{6} x + a b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/210*(105*(9*B*a^4 - 7*A*a^3*b + (9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(-a/b)*log((b*x - 2*b*sqrt(x)*sqrt(-a/b)
- a)/(b*x + a)) - 2*(30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b - 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 -
7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b^6*x + a*b^5), 1/105*(105*(9*B*a^4 - 7*A*a^3*b + (
9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(a/b)*arctan(b*sqrt(x)*sqrt(a/b)/a) + (30*B*b^4*x^4 - 945*B*a^4 + 735*A*a^3*b
- 6*(9*B*a*b^3 - 7*A*b^4)*x^3 + 14*(9*B*a^2*b^2 - 7*A*a*b^3)*x^2 - 70*(9*B*a^3*b - 7*A*a^2*b^2)*x)*sqrt(x))/(b
^6*x + a*b^5)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (143) = 286\).
time = 129.41, size = 986, normalized size = 6.40 \begin {gather*} \begin {cases} \tilde {\infty } \left (\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {\frac {2 A x^{\frac {9}{2}}}{9} + \frac {2 B x^{\frac {11}{2}}}{11}}{a^{2}} & \text {for}\: b = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{b^{2}} & \text {for}\: a = 0 \\- \frac {735 A a^{4} b \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{4} b \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {1470 A a^{3} b^{2} \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {735 A a^{3} b^{2} x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {980 A a^{2} b^{3} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {196 A a b^{4} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {84 A b^{5} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{5} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{5} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1890 B a^{4} b \sqrt {x} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {945 B a^{4} b x \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {945 B a^{4} b x \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {1260 B a^{3} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {252 B a^{2} b^{3} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} - \frac {108 B a b^{4} x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} + \frac {60 B b^{5} x^{\frac {9}{2}} \sqrt {- \frac {a}{b}}}{210 a b^{6} \sqrt {- \frac {a}{b}} + 210 b^{7} x \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*(2*A*x**(5/2)/5 + 2*B*x**(7/2)/7), Eq(a, 0) & Eq(b, 0)), ((2*A*x**(9/2)/9 + 2*B*x**(11/2)/11)/a
**2, Eq(b, 0)), ((2*A*x**(5/2)/5 + 2*B*x**(7/2)/7)/b**2, Eq(a, 0)), (-735*A*a**4*b*log(sqrt(x) - sqrt(-a/b))/(
210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 735*A*a**4*b*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b)
+ 210*b**7*x*sqrt(-a/b)) + 1470*A*a**3*b**2*sqrt(x)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)
) - 735*A*a**3*b**2*x*log(sqrt(x) - sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 735*A*a**3*b
**2*x*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 980*A*a**2*b**3*x**(3/2)*sqr
t(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 196*A*a*b**4*x**(5/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a
/b) + 210*b**7*x*sqrt(-a/b)) + 84*A*b**5*x**(7/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) +
945*B*a**5*log(sqrt(x) - sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 945*B*a**5*log(sqrt(x)
+ sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 1890*B*a**4*b*sqrt(x)*sqrt(-a/b)/(210*a*b**6*
sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 945*B*a**4*b*x*log(sqrt(x) - sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b*
*7*x*sqrt(-a/b)) - 945*B*a**4*b*x*log(sqrt(x) + sqrt(-a/b))/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) -
1260*B*a**3*b**2*x**(3/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 252*B*a**2*b**3*x**(5/2
)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/b)) - 108*B*a*b**4*x**(7/2)*sqrt(-a/b)/(210*a*b**6*sq
rt(-a/b) + 210*b**7*x*sqrt(-a/b)) + 60*B*b**5*x**(9/2)*sqrt(-a/b)/(210*a*b**6*sqrt(-a/b) + 210*b**7*x*sqrt(-a/
b)), True))

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Giac [A]
time = 1.30, size = 146, normalized size = 0.95 \begin {gather*} \frac {{\left (9 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} b^{5}} - \frac {B a^{4} \sqrt {x} - A a^{3} b \sqrt {x}}{{\left (b x + a\right )} b^{5}} + \frac {2 \, {\left (15 \, B b^{12} x^{\frac {7}{2}} - 42 \, B a b^{11} x^{\frac {5}{2}} + 21 \, A b^{12} x^{\frac {5}{2}} + 105 \, B a^{2} b^{10} x^{\frac {3}{2}} - 70 \, A a b^{11} x^{\frac {3}{2}} - 420 \, B a^{3} b^{9} \sqrt {x} + 315 \, A a^{2} b^{10} \sqrt {x}\right )}}{105 \, b^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*B*a^4 - 7*A*a^3*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b^5) - (B*a^4*sqrt(x) - A*a^3*b*sqrt(x))/((b*x +
a)*b^5) + 2/105*(15*B*b^12*x^(7/2) - 42*B*a*b^11*x^(5/2) + 21*A*b^12*x^(5/2) + 105*B*a^2*b^10*x^(3/2) - 70*A*a
*b^11*x^(3/2) - 420*B*a^3*b^9*sqrt(x) + 315*A*a^2*b^10*sqrt(x))/b^14

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Mupad [B]
time = 0.07, size = 209, normalized size = 1.36 \begin {gather*} \sqrt {x}\,\left (\frac {2\,a\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b}+\frac {2\,B\,a^2}{b^4}\right )}{b}-\frac {a^2\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{b^2}\right )+x^{5/2}\,\left (\frac {2\,A}{5\,b^2}-\frac {4\,B\,a}{5\,b^3}\right )-x^{3/2}\,\left (\frac {2\,a\,\left (\frac {2\,A}{b^2}-\frac {4\,B\,a}{b^3}\right )}{3\,b}+\frac {2\,B\,a^2}{3\,b^4}\right )+\frac {2\,B\,x^{7/2}}{7\,b^2}-\frac {\sqrt {x}\,\left (B\,a^4-A\,a^3\,b\right )}{x\,b^6+a\,b^5}+\frac {a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,\sqrt {x}\,\left (7\,A\,b-9\,B\,a\right )}{9\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-9\,B\,a\right )}{b^{11/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)*((2*a*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/b + (2*B*a^2)/b^4))/b - (a^2*((2*A)/b^2 - (4*B*a)/b^3))/b^2) +
x^(5/2)*((2*A)/(5*b^2) - (4*B*a)/(5*b^3)) - x^(3/2)*((2*a*((2*A)/b^2 - (4*B*a)/b^3))/(3*b) + (2*B*a^2)/(3*b^4)
) + (2*B*x^(7/2))/(7*b^2) - (x^(1/2)*(B*a^4 - A*a^3*b))/(a*b^5 + b^6*x) + (a^(5/2)*atan((a^(5/2)*b^(1/2)*x^(1/
2)*(7*A*b - 9*B*a))/(9*B*a^4 - 7*A*a^3*b))*(7*A*b - 9*B*a))/b^(11/2)

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