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

Optimal. Leaf size=320 \[ \frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]

[Out]

-(x^(5/2)*(A + B*x))/(4*c*(a + c*x^2)^2) - (Sqrt[x]*(5*A + 7*B*x))/(16*c^2*(a + c*x^2)) - ((21*Sqrt[a]*B + 5*A
*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B + 5*A*
Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B - 5*A*S
qrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4)) - ((21*Sqrt[
a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4))

________________________________________________________________________________________

Rubi [A]  time = 0.306181, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {819, 827, 1168, 1162, 617, 204, 1165, 628} \[ \frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x^(5/2)*(A + B*x))/(4*c*(a + c*x^2)^2) - (Sqrt[x]*(5*A + 7*B*x))/(16*c^2*(a + c*x^2)) - ((21*Sqrt[a]*B + 5*A
*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B + 5*A*
Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(32*Sqrt[2]*a^(3/4)*c^(11/4)) + ((21*Sqrt[a]*B - 5*A*S
qrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4)) - ((21*Sqrt[
a]*B - 5*A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/(64*Sqrt[2]*a^(3/4)*c^(11/4))

Rule 819

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 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{x^{7/2} (A+B x)}{\left (a+c x^2\right )^3} \, dx &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}+\frac{\int \frac{x^{3/2} \left (\frac{5 a A}{2}+\frac{7 a B x}{2}\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac{\int \frac{\frac{5 a^2 A}{4}+\frac{21}{4} a^2 B x}{\sqrt{x} \left (a+c x^2\right )} \, dx}{8 a^2 c^2}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{5 a^2 A}{4}+\frac{21}{4} a^2 B x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{4 a^2 c^2}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{\left (21 B-\frac{5 A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}-c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^3}+\frac{\left (21 B+\frac{5 A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a} \sqrt{c}+c x^2}{a+c x^4} \, dx,x,\sqrt{x}\right )}{32 c^3}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac{\left (21 B+\frac{5 A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 c^3}+\frac{\left (21 B+\frac{5 A \sqrt{c}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt{x}\right )}{64 c^3}+\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac{\sqrt{a}}{\sqrt{c}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac{\sqrt{a}}{\sqrt{c}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}+\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}\\ &=-\frac{x^{5/2} (A+B x)}{4 c \left (a+c x^2\right )^2}-\frac{\sqrt{x} (5 A+7 B x)}{16 c^2 \left (a+c x^2\right )}-\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B+5 A \sqrt{c}\right ) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{32 \sqrt{2} a^{3/4} c^{11/4}}+\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}-\frac{\left (21 \sqrt{a} B-5 A \sqrt{c}\right ) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{c} x\right )}{64 \sqrt{2} a^{3/4} c^{11/4}}\\ \end{align*}

Mathematica [A]  time = 0.55415, size = 385, normalized size = 1.2 \[ \frac{-\frac{5 \sqrt{2} a^{5/4} A \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{9/4}}+\frac{5 \sqrt{2} a^{5/4} A \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{9/4}}-\frac{10 \sqrt{2} a^{5/4} A \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{c^{9/4}}+\frac{10 \sqrt{2} a^{5/4} A \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{c^{9/4}}-\frac{40 a A \sqrt{x}}{c^2}-\frac{8 A x^{9/2}}{a+c x^2}+\frac{32 a A x^{9/2}}{\left (a+c x^2\right )^2}-\frac{56 a B x^{3/2}}{c^2}+\frac{84 (-a)^{7/4} B \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{11/4}}+\frac{84 (-a)^{3/4} a B \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{11/4}}-\frac{24 B x^{11/2}}{a+c x^2}+\frac{32 a B x^{11/2}}{\left (a+c x^2\right )^2}+\frac{8 A x^{5/2}}{c}+\frac{24 B x^{7/2}}{c}}{128 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-40*a*A*Sqrt[x])/c^2 - (56*a*B*x^(3/2))/c^2 + (8*A*x^(5/2))/c + (24*B*x^(7/2))/c + (32*a*A*x^(9/2))/(a + c*x
^2)^2 + (32*a*B*x^(11/2))/(a + c*x^2)^2 - (8*A*x^(9/2))/(a + c*x^2) - (24*B*x^(11/2))/(a + c*x^2) - (10*Sqrt[2
]*a^(5/4)*A*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(9/4) + (10*Sqrt[2]*a^(5/4)*A*ArcTan[1 + (Sqrt[2]
*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(9/4) + (84*(-a)^(7/4)*B*ArcTan[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(11/4) + (84*(-
a)^(3/4)*a*B*ArcTanh[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(11/4) - (5*Sqrt[2]*a^(5/4)*A*Log[Sqrt[a] - Sqrt[2]*a^(1
/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c^(9/4) + (5*Sqrt[2]*a^(5/4)*A*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x]
+ Sqrt[c]*x])/c^(9/4))/(128*a^2)

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 327, normalized size = 1. \begin{align*} 2\,{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{11\,B{x}^{7/2}}{32\,c}}-{\frac{9\,A{x}^{5/2}}{32\,c}}-{\frac{7\,aB{x}^{3/2}}{32\,{c}^{2}}}-{\frac{5\,aA\sqrt{x}}{32\,{c}^{2}}} \right ) }+{\frac{5\,A\sqrt{2}}{128\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,A\sqrt{2}}{64\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{5\,A\sqrt{2}}{64\,a{c}^{2}}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{21\,B\sqrt{2}}{128\,{c}^{3}}\ln \left ({ \left ( x-\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ( x+\sqrt [4]{{\frac{a}{c}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,B\sqrt{2}}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{21\,B\sqrt{2}}{64\,{c}^{3}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-11/32*B*x^(7/2)/c-9/32/c*A*x^(5/2)-7/32*a*B/c^2*x^(3/2)-5/32*a*A/c^2*x^(1/2))/(c*x^2+a)^2+5/128/c^2*A*(a/c
)^(1/4)/a*2^(1/2)*ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+
5/64/c^2*A*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+5/64/c^2*A*(a/c)^(1/4)/a*2^(1/2)*arctan
(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+21/128/c^3*B/(a/c)^(1/4)*2^(1/2)*ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)
)/(x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+21/64/c^3*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1
/2)+1)+21/64/c^3*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.54298, size = 2326, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*((c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt(-(a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/
(a^3*c^11)) + 210*A*B)/(a*c^5))*log(-(194481*B^4*a^2 - 625*A^4*c^2)*sqrt(x) + (21*B*a^3*c^8*sqrt(-(194481*B^4*
a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 2205*A*B^2*a^2*c^3 + 125*A^3*a*c^4)*sqrt(-(a*c^5*sqrt(-(1
94481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B)/(a*c^5))) - (c^4*x^4 + 2*a*c^3*x^2 + a
^2*c^2)*sqrt(-(a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B)/(a*c^5))*
log(-(194481*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (21*B*a^3*c^8*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^
4*c^2)/(a^3*c^11)) - 2205*A*B^2*a^2*c^3 + 125*A^3*a*c^4)*sqrt(-(a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*
c + 625*A^4*c^2)/(a^3*c^11)) + 210*A*B)/(a*c^5))) - (c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt((a*c^5*sqrt(-(19448
1*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 210*A*B)/(a*c^5))*log(-(194481*B^4*a^2 - 625*A^4*c^
2)*sqrt(x) + (21*B*a^3*c^8*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 2205*A*B^2*a
^2*c^3 - 125*A^3*a*c^4)*sqrt((a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 210
*A*B)/(a*c^5))) + (c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)*sqrt((a*c^5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 62
5*A^4*c^2)/(a^3*c^11)) - 210*A*B)/(a*c^5))*log(-(194481*B^4*a^2 - 625*A^4*c^2)*sqrt(x) - (21*B*a^3*c^8*sqrt(-(
194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) + 2205*A*B^2*a^2*c^3 - 125*A^3*a*c^4)*sqrt((a*c^
5*sqrt(-(194481*B^4*a^2 - 22050*A^2*B^2*a*c + 625*A^4*c^2)/(a^3*c^11)) - 210*A*B)/(a*c^5))) - 4*(11*B*c*x^3 +
9*A*c*x^2 + 7*B*a*x + 5*A*a)*sqrt(x))/(c^4*x^4 + 2*a*c^3*x^2 + a^2*c^2)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(7/2)*(B*x+A)/(c*x**2+a)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.35261, size = 396, normalized size = 1.24 \begin{align*} -\frac{11 \, B c x^{\frac{7}{2}} + 9 \, A c x^{\frac{5}{2}} + 7 \, B a x^{\frac{3}{2}} + 5 \, A a \sqrt{x}}{16 \,{\left (c x^{2} + a\right )}^{2} c^{2}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a c^{5}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{64 \, a c^{5}} + \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a c^{5}} - \frac{\sqrt{2}{\left (5 \, \left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 21 \, \left (a c^{3}\right )^{\frac{3}{4}} B\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{c}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{c}}\right )}{128 \, a c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(11*B*c*x^(7/2) + 9*A*c*x^(5/2) + 7*B*a*x^(3/2) + 5*A*a*sqrt(x))/((c*x^2 + a)^2*c^2) + 1/64*sqrt(2)*(5*(
a*c^3)^(1/4)*A*c^2 + 21*(a*c^3)^(3/4)*B)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a*
c^5) + 1/64*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 + 21*(a*c^3)^(3/4)*B)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*
sqrt(x))/(a/c)^(1/4))/(a*c^5) + 1/128*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 - 21*(a*c^3)^(3/4)*B)*log(sqrt(2)*sqrt(x)
*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^5) - 1/128*sqrt(2)*(5*(a*c^3)^(1/4)*A*c^2 - 21*(a*c^3)^(3/4)*B)*log(-sqrt(2
)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^5)