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

Optimal. Leaf size=289 \[ \frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{2 c \left (a+c x^2\right )} \]

[Out]

-(Sqrt[x]*(A + B*x))/(2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4)) - ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4))

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Rubi [A]  time = 0.244181, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B-A \sqrt{c}\right ) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{8 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}+\frac{\left (3 \sqrt{a} B+A \sqrt{c}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{3/4} c^{7/4}}-\frac{\sqrt{x} (A+B x)}{2 c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[x]*(A + B*x))/(2*c*(a + c*x^2)) - ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/
4)])/(4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B + A*Sqrt[c])*ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/(
4*Sqrt[2]*a^(3/4)*c^(7/4)) + ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c
]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4)) - ((3*Sqrt[a]*B - A*Sqrt[c])*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] +
Sqrt[c]*x])/(8*Sqrt[2]*a^(3/4)*c^(7/4))

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

Mathematica [A]  time = 0.254818, size = 323, normalized size = 1.12 \[ \frac{-\frac{\sqrt{2} \sqrt [4]{a} A \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{5/4}}+\frac{\sqrt{2} \sqrt [4]{a} A \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} \sqrt{x}+\sqrt{a}+\sqrt{c} x\right )}{c^{5/4}}-\frac{2 \sqrt{2} \sqrt [4]{a} A \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )}{c^{5/4}}+\frac{2 \sqrt{2} \sqrt [4]{a} A \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{c^{5/4}}+\frac{8 A x^{5/2}}{a+c x^2}-\frac{12 (-a)^{3/4} B \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac{12 (-a)^{3/4} B \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{-a}}\right )}{c^{7/4}}+\frac{8 B x^{7/2}}{a+c x^2}-\frac{8 A \sqrt{x}}{c}-\frac{8 B x^{3/2}}{c}}{16 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*A*Sqrt[x])/c - (8*B*x^(3/2))/c + (8*A*x^(5/2))/(a + c*x^2) + (8*B*x^(7/2))/(a + c*x^2) - (2*Sqrt[2]*a^(1/
4)*A*ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)])/c^(5/4) + (2*Sqrt[2]*a^(1/4)*A*ArcTan[1 + (Sqrt[2]*c^(1/4)
*Sqrt[x])/a^(1/4)])/c^(5/4) - (12*(-a)^(3/4)*B*ArcTan[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(7/4) + (12*(-a)^(3/4)*
B*ArcTanh[(c^(1/4)*Sqrt[x])/(-a)^(1/4)])/c^(7/4) - (Sqrt[2]*a^(1/4)*A*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*Sq
rt[x] + Sqrt[c]*x])/c^(5/4) + (Sqrt[2]*a^(1/4)*A*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x])/c
^(5/4))/(16*a)

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Maple [A]  time = 0.012, size = 307, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{c{x}^{2}+a} \left ( -1/4\,{\frac{B{x}^{3/2}}{c}}-1/4\,{\frac{A\sqrt{x}}{c}} \right ) }+{\frac{A\sqrt{2}}{16\,ac}\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{A\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{A\sqrt{2}}{8\,ac}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) }+{\frac{3\,B\sqrt{2}}{16\,{c}^{2}}\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{3\,B\sqrt{2}}{8\,{c}^{2}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,B\sqrt{2}}{8\,{c}^{2}}\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^(3/2)*(B*x+A)/(c*x^2+a)^2,x)

[Out]

2*(-1/4*B*x^(3/2)/c-1/4*A*x^(1/2)/c)/(c*x^2+a)+1/16/c*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)))+1/8/c*A*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)
^(1/4)*x^(1/2)+1)+1/8/c*A*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)+3/16/c^2*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)))+3/8/c^2*B/(
a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+3/8/c^2*B/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1
/4)*x^(1/2)-1)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56172, size = 1856, normalized size = 6.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((c^2*x^2 + a*c)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3))*l
og(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) + (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 9
*A*B^2*a^2*c^2 + A^3*a*c^3)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*
c^3))) - (c^2*x^2 + a*c)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3
))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) - (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7))
- 9*A*B^2*a^2*c^2 + A^3*a*c^3)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)
/(a*c^3))) - (c^2*x^2 + a*c)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*
c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) + (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^
7)) + 9*A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*
B)/(a*c^3))) + (c^2*x^2 + a*c)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(
a*c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) - (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*
c^7)) + 9*A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*
A*B)/(a*c^3))) - 4*(B*x + A)*sqrt(x))/(c^2*x^2 + a*c)

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

[Out]

Timed out

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Giac [A]  time = 1.37057, size = 366, normalized size = 1.27 \begin{align*} -\frac{B x^{\frac{3}{2}} + A \sqrt{x}}{2 \,{\left (c x^{2} + a\right )} c} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 3 \, \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 )}{8 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} + 3 \, \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 )}{8 \, a c^{4}} + \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 3 \, \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 )}{16 \, a c^{4}} - \frac{\sqrt{2}{\left (\left (a c^{3}\right )^{\frac{1}{4}} A c^{2} - 3 \, \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 )}{16 \, a c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(B*x^(3/2) + A*sqrt(x))/((c*x^2 + a)*c) + 1/8*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + 3*(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^4) + 1/8*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + 3*(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^4) + 1/16*sqrt(2)*((a*c^3)^
(1/4)*A*c^2 - 3*(a*c^3)^(3/4)*B)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^4) - 1/16*sqrt(2)*((a*c
^3)^(1/4)*A*c^2 - 3*(a*c^3)^(3/4)*B)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^4)