3.1.47 \(\int \frac {x^7 (A+B x+C x^2)}{(a+b x^2)^{9/2}} \, dx\) [47]

Optimal. Leaf size=213 \[ -\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \]

[Out]

-1/7*x^7*(a*B-(A*b-C*a)*x)/a/b/(b*x^2+a)^(7/2)-1/35*x^5*(7*a*B-(A*b-8*C*a)*x)/a/b^2/(b*x^2+a)^(5/2)-1/105*x^3*
(35*a*B-6*(A*b-8*C*a)*x)/a/b^3/(b*x^2+a)^(3/2)+B*arctanh(x*b^(1/2)/(b*x^2+a)^(1/2))/b^(9/2)-1/35*x*(35*a*B-8*(
A*b-8*C*a)*x)/a/b^4/(b*x^2+a)^(1/2)-16/35*(A*b-8*C*a)*(b*x^2+a)^(1/2)/a/b^5

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Rubi [A]
time = 0.23, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1818, 833, 655, 223, 212} \begin {gather*} -\frac {16 \sqrt {a+b x^2} (A b-8 a C)}{35 a b^5}-\frac {x (35 a B-8 x (A b-8 a C))}{35 a b^4 \sqrt {a+b x^2}}-\frac {x^3 (35 a B-6 x (A b-8 a C))}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x^5 (7 a B-x (A b-8 a C))}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^7 (a B-x (A b-a C))}{7 a b \left (a+b x^2\right )^{7/2}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

-1/7*(x^7*(a*B - (A*b - a*C)*x))/(a*b*(a + b*x^2)^(7/2)) - (x^5*(7*a*B - (A*b - 8*a*C)*x))/(35*a*b^2*(a + b*x^
2)^(5/2)) - (x^3*(35*a*B - 6*(A*b - 8*a*C)*x))/(105*a*b^3*(a + b*x^2)^(3/2)) - (x*(35*a*B - 8*(A*b - 8*a*C)*x)
)/(35*a*b^4*Sqrt[a + b*x^2]) - (16*(A*b - 8*a*C)*Sqrt[a + b*x^2])/(35*a*b^5) + (B*ArcTanh[(Sqrt[b]*x)/Sqrt[a +
b*x^2]])/b^(9/2)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 655

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[e*((a + c*x^2)^(p + 1)/(2*c*(p + 1))),
x] + Dist[d, Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[p, -1]

Rule 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
- 1)*(a + c*x^2)^(p + 1)*((a*(e*f + d*g) - (c*d*f - a*e*g)*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 1818

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x
^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x
], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Dist[c/(2*a*b*(p + 1)), Int[
(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /;
FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {x^7 \left (A+B x+C x^2\right )}{\left (a+b x^2\right )^{9/2}} \, dx &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {\int \frac {x^6 (-7 a B+(A b-8 a C) x)}{\left (a+b x^2\right )^{7/2}} \, dx}{7 a b}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {\int \frac {x^4 \left (-35 a^2 B+6 a (A b-8 a C) x\right )}{\left (a+b x^2\right )^{5/2}} \, dx}{35 a^2 b^2}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {\int \frac {x^2 \left (-105 a^3 B+24 a^2 (A b-8 a C) x\right )}{\left (a+b x^2\right )^{3/2}} \, dx}{105 a^3 b^3}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {\int \frac {-105 a^4 B+48 a^3 (A b-8 a C) x}{\sqrt {a+b x^2}} \, dx}{105 a^4 b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b^4}\\ &=-\frac {x^7 (a B-(A b-a C) x)}{7 a b \left (a+b x^2\right )^{7/2}}-\frac {x^5 (7 a B-(A b-8 a C) x)}{35 a b^2 \left (a+b x^2\right )^{5/2}}-\frac {x^3 (35 a B-6 (A b-8 a C) x)}{105 a b^3 \left (a+b x^2\right )^{3/2}}-\frac {x (35 a B-8 (A b-8 a C) x)}{35 a b^4 \sqrt {a+b x^2}}-\frac {16 (A b-8 a C) \sqrt {a+b x^2}}{35 a b^5}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 1.15, size = 156, normalized size = 0.73 \begin {gather*} \frac {384 a^4 C-3 a^3 b (16 A+7 x (5 B-64 C x))+14 a^2 b^2 x^2 (-12 A+5 x (-5 B+24 C x))+14 a b^3 x^4 (-15 A+x (-29 B+60 C x))+b^4 x^6 (-105 A+x (-176 B+105 C x))-105 \sqrt {b} B \left (a+b x^2\right )^{7/2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{105 b^5 \left (a+b x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x]

[Out]

(384*a^4*C - 3*a^3*b*(16*A + 7*x*(5*B - 64*C*x)) + 14*a^2*b^2*x^2*(-12*A + 5*x*(-5*B + 24*C*x)) + 14*a*b^3*x^4
*(-15*A + x*(-29*B + 60*C*x)) + b^4*x^6*(-105*A + x*(-176*B + 105*C*x)) - 105*Sqrt[b]*B*(a + b*x^2)^(7/2)*Log[
-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(105*b^5*(a + b*x^2)^(7/2))

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Maple [A]
time = 0.17, size = 295, normalized size = 1.38

method	result	size
default	\(C \left (\frac {x^{8}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {8 a \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )}{b}\right )+B \left (-\frac {x^{7}}{7 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {-\frac {x^{5}}{5 b \left (b \,x^{2}+a \right )^{\frac {5}{2}}}+\frac {-\frac {x^{3}}{3 b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}+\frac {-\frac {x}{b \sqrt {b \,x^{2}+a}}+\frac {\ln \left (x \sqrt {b}+\sqrt {b \,x^{2}+a}\right )}{b^{\frac {3}{2}}}}{b}}{b}}{b}\right )+A \left (-\frac {x^{6}}{b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {6 a \left (-\frac {x^{4}}{3 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}+\frac {4 a \left (-\frac {x^{2}}{5 b \left (b \,x^{2}+a \right )^{\frac {7}{2}}}-\frac {2 a}{35 b^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}\right )}{3 b}\right )}{b}\right )\)	\(295\)
risch	\(\text {Expression too large to display}\)	\(2131\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x,method=_RETURNVERBOSE)

[Out]

C*(x^8/b/(b*x^2+a)^(7/2)-8*a/b*(-x^6/b/(b*x^2+a)^(7/2)+6*a/b*(-1/3*x^4/b/(b*x^2+a)^(7/2)+4/3*a/b*(-1/5*x^2/b/(
b*x^2+a)^(7/2)-2/35*a/b^2/(b*x^2+a)^(7/2)))))+B*(-1/7*x^7/b/(b*x^2+a)^(7/2)+1/b*(-1/5*x^5/b/(b*x^2+a)^(5/2)+1/
b*(-1/3*x^3/b/(b*x^2+a)^(3/2)+1/b*(-x/b/(b*x^2+a)^(1/2)+1/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))))))+A*(-x^6/b/
(b*x^2+a)^(7/2)+6*a/b*(-1/3*x^4/b/(b*x^2+a)^(7/2)+4/3*a/b*(-1/5*x^2/b/(b*x^2+a)^(7/2)-2/35*a/b^2/(b*x^2+a)^(7/
2))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (189) = 378\).
time = 0.30, size = 435, normalized size = 2.04 \begin {gather*} \frac {C x^{8}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {1}{35} \, {\left (\frac {35 \, x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} + \frac {70 \, a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} + \frac {56 \, a^{2} x^{2}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {16 \, a^{3}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}}\right )} B x + \frac {8 \, C a x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {A x^{6}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b} - \frac {B x {\left (\frac {15 \, x^{4}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b} + \frac {20 \, a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{2}} + \frac {8 \, a^{2}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}}\right )}}{15 \, b} - \frac {B x {\left (\frac {3 \, x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {2 \, a}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}}\right )}}{3 \, b^{2}} + \frac {16 \, C a^{2} x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} - \frac {2 \, A a x^{4}}{{\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{2}} - \frac {B a x^{3}}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{3}} + \frac {64 \, C a^{3} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} - \frac {8 \, A a^{2} x^{2}}{5 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{3}} + \frac {139 \, B x}{105 \, \sqrt {b x^{2} + a} b^{4}} + \frac {17 \, B a x}{105 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{4}} - \frac {29 \, B a^{2} x}{35 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} b^{4}} + \frac {B \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {9}{2}}} + \frac {128 \, C a^{4}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{5}} - \frac {16 \, A a^{3}}{35 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

C*x^8/((b*x^2 + a)^(7/2)*b) - 1/35*(35*x^6/((b*x^2 + a)^(7/2)*b) + 70*a*x^4/((b*x^2 + a)^(7/2)*b^2) + 56*a^2*x
^2/((b*x^2 + a)^(7/2)*b^3) + 16*a^3/((b*x^2 + a)^(7/2)*b^4))*B*x + 8*C*a*x^6/((b*x^2 + a)^(7/2)*b^2) - A*x^6/(
(b*x^2 + a)^(7/2)*b) - 1/15*B*x*(15*x^4/((b*x^2 + a)^(5/2)*b) + 20*a*x^2/((b*x^2 + a)^(5/2)*b^2) + 8*a^2/((b*x
^2 + a)^(5/2)*b^3))/b - 1/3*B*x*(3*x^2/((b*x^2 + a)^(3/2)*b) + 2*a/((b*x^2 + a)^(3/2)*b^2))/b^2 + 16*C*a^2*x^4
/((b*x^2 + a)^(7/2)*b^3) - 2*A*a*x^4/((b*x^2 + a)^(7/2)*b^2) - B*a*x^3/((b*x^2 + a)^(5/2)*b^3) + 64/5*C*a^3*x^
2/((b*x^2 + a)^(7/2)*b^4) - 8/5*A*a^2*x^2/((b*x^2 + a)^(7/2)*b^3) + 139/105*B*x/(sqrt(b*x^2 + a)*b^4) + 17/105
*B*a*x/((b*x^2 + a)^(3/2)*b^4) - 29/35*B*a^2*x/((b*x^2 + a)^(5/2)*b^4) + B*arcsinh(b*x/sqrt(a*b))/b^(9/2) + 12
8/35*C*a^4/((b*x^2 + a)^(7/2)*b^5) - 16/35*A*a^3/((b*x^2 + a)^(7/2)*b^4)

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Fricas [A]
time = 3.53, size = 522, normalized size = 2.45 \begin {gather*} \left [\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) + 2 \, {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{210 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, -\frac {105 \, {\left (B b^{4} x^{8} + 4 \, B a b^{3} x^{6} + 6 \, B a^{2} b^{2} x^{4} + 4 \, B a^{3} b x^{2} + B a^{4}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (105 \, C b^{4} x^{8} - 176 \, B b^{4} x^{7} - 406 \, B a b^{3} x^{5} - 350 \, B a^{2} b^{2} x^{3} + 105 \, {\left (8 \, C a b^{3} - A b^{4}\right )} x^{6} - 105 \, B a^{3} b x + 384 \, C a^{4} - 48 \, A a^{3} b + 210 \, {\left (8 \, C a^{2} b^{2} - A a b^{3}\right )} x^{4} + 168 \, {\left (8 \, C a^{3} b - A a^{2} b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{105 \, {\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

[1/210*(105*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*sqrt(b)*log(-2*b*x^2 - 2*sqr
t(b*x^2 + a)*sqrt(b)*x - a) + 2*(105*C*b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*
C*a*b^3 - A*b^4)*x^6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3
*b - A*a^2*b^2)*x^2)*sqrt(b*x^2 + a))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5), -1/10
5*(105*(B*b^4*x^8 + 4*B*a*b^3*x^6 + 6*B*a^2*b^2*x^4 + 4*B*a^3*b*x^2 + B*a^4)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b
*x^2 + a)) - (105*C*b^4*x^8 - 176*B*b^4*x^7 - 406*B*a*b^3*x^5 - 350*B*a^2*b^2*x^3 + 105*(8*C*a*b^3 - A*b^4)*x^
6 - 105*B*a^3*b*x + 384*C*a^4 - 48*A*a^3*b + 210*(8*C*a^2*b^2 - A*a*b^3)*x^4 + 168*(8*C*a^3*b - A*a^2*b^2)*x^2
)*sqrt(b*x^2 + a))/(b^9*x^8 + 4*a*b^8*x^6 + 6*a^2*b^7*x^4 + 4*a^3*b^6*x^2 + a^4*b^5)]

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Sympy [A]
time = 103.21, size = 3806, normalized size = 17.87 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(C*x**2+B*x+A)/(b*x**2+a)**(9/2),x)

[Out]

A*Piecewise((-16*a**3/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*s
qrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 56*a**2*b*x**2/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b*
*5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)) - 70*a*b**2*x**4/
(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2) + 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b
**7*x**6*sqrt(a + b*x**2)) - 35*b**3*x**6/(35*a**3*b**4*sqrt(a + b*x**2) + 105*a**2*b**5*x**2*sqrt(a + b*x**2)
+ 105*a*b**6*x**4*sqrt(a + b*x**2) + 35*b**7*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**8/(8*a**(9/2)), True)) +
B*(105*a**(205/2)*b**45*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/
a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) +
2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*
a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 630*a**
(203/2)*b**46*x**2*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) +
630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*
a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(1
95/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(201
/2)*b**47*x**4*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*
a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(
199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2
)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 2100*a**(199/2)*
b**48*x**6*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(
203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/
2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b*
*(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 1575*a**(197/2)*b**4
9*x**8*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/
2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b
**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(10
9/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 630*a**(195/2)*b**50*x**
10*sqrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b
**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(1
05/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)
*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) + 105*a**(193/2)*b**51*x**12*s
qrt(1 + b*x**2/a)*asinh(sqrt(b)*x/sqrt(a))/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(1
01/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2
)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**
10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 105*a**102*b**(91/2)*x/(105*a**(
205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(
103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/
2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x*
*12*sqrt(1 + b*x**2/a)) - 665*a**101*b**(93/2)*x**3/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203
/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a**(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*
b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(197/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(1
09/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 1771*a**100*b**(95/2)*x
**5/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a) + 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**2/a) + 1575*a*
*(201/2)*b**(103/2)*x**4*sqrt(1 + b*x**2/a) + 2100*a**(199/2)*b**(105/2)*x**6*sqrt(1 + b*x**2/a) + 1575*a**(19
7/2)*b**(107/2)*x**8*sqrt(1 + b*x**2/a) + 630*a**(195/2)*b**(109/2)*x**10*sqrt(1 + b*x**2/a) + 105*a**(193/2)*
b**(111/2)*x**12*sqrt(1 + b*x**2/a)) - 2549*a**99*b**(97/2)*x**7/(105*a**(205/2)*b**(99/2)*sqrt(1 + b*x**2/a)
+ 630*a**(203/2)*b**(101/2)*x**2*sqrt(1 + b*x**...

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Giac [A]
time = 0.73, size = 204, normalized size = 0.96 \begin {gather*} \frac {{\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac {105 \, C x}{b} - \frac {176 \, B}{b}\right )} x + \frac {105 \, {\left (8 \, C a^{4} b^{7} - A a^{3} b^{8}\right )}}{a^{3} b^{9}}\right )} x - \frac {406 \, B a}{b^{2}}\right )} x + \frac {210 \, {\left (8 \, C a^{5} b^{6} - A a^{4} b^{7}\right )}}{a^{3} b^{9}}\right )} x - \frac {350 \, B a^{2}}{b^{3}}\right )} x + \frac {168 \, {\left (8 \, C a^{6} b^{5} - A a^{5} b^{6}\right )}}{a^{3} b^{9}}\right )} x - \frac {105 \, B a^{3}}{b^{4}}\right )} x + \frac {48 \, {\left (8 \, C a^{7} b^{4} - A a^{6} b^{5}\right )}}{a^{3} b^{9}}}{105 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}}} - \frac {B \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(C*x^2+B*x+A)/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/105*((((((((105*C*x/b - 176*B/b)*x + 105*(8*C*a^4*b^7 - A*a^3*b^8)/(a^3*b^9))*x - 406*B*a/b^2)*x + 210*(8*C*
a^5*b^6 - A*a^4*b^7)/(a^3*b^9))*x - 350*B*a^2/b^3)*x + 168*(8*C*a^6*b^5 - A*a^5*b^6)/(a^3*b^9))*x - 105*B*a^3/
b^4)*x + 48*(8*C*a^7*b^4 - A*a^6*b^5)/(a^3*b^9))/(b*x^2 + a)^(7/2) - B*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/
b^(9/2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^7\,\left (C\,x^2+B\,x+A\right )}{{\left (b\,x^2+a\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2),x)

[Out]

int((x^7*(A + B*x + C*x^2))/(a + b*x^2)^(9/2), x)

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