3.460 \(\int \frac{(c+d x^2)^3}{x^{9/2} (a+b x^2)^2} \, dx\)

Optimal. Leaf size=376 \[ \frac{c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}-\frac{(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]

[Out]

-(c^2*(11*b*c - 7*a*d))/(14*a^2*b*x^(7/2)) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2*d^2))/(6*a^3*b*x^(3/2)) + ((b
*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2)) - ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4))

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Rubi [A]  time = 0.415759, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {466, 468, 570, 211, 1165, 628, 1162, 617, 204} \[ \frac{c \left (6 a^2 d^2-21 a b c d+11 b^2 c^2\right )}{6 a^3 b x^{3/2}}-\frac{(b c-a d)^2 (a d+11 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (a d+11 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{7/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

-(c^2*(11*b*c - 7*a*d))/(14*a^2*b*x^(7/2)) + (c*(11*b^2*c^2 - 21*a*b*c*d + 6*a^2*d^2))/(6*a^3*b*x^(3/2)) + ((b
*c - a*d)*(c + d*x^2)^2)/(2*a*b*x^(7/2)*(a + b*x^2)) - ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/
4)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(15/4)*b^(5/4)) - ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)
*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4)) + ((b*c - a*d)^2*(11*b*c + a*d)*Log[Sqrt[a] + Sqrt
[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(15/4)*b^(5/4))

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 468

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[((c*b -
a*d)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1))/(a*b*e*n*(p + 1)), x] + Dist[1/(a*b*n*(p + 1)), I
nt[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c*b - a*d)*(m + 1)) + d*(c*b*n*(p
+ 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

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]

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])

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^3}{x^{9/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (c+d x^4\right )^3}{x^8 \left (a+b x^4\right )^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\left (c+d x^4\right ) \left (-c (11 b c-7 a d)-d (3 b c+a d) x^4\right )}{x^8 \left (a+b x^4\right )} \, dx,x,\sqrt{x}\right )}{2 a b}\\ &=\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{c^2 (-11 b c+7 a d)}{a x^8}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{a^2 x^4}-\frac{(-b c+a d)^2 (11 b c+a d)}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 a b}\\ &=-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^3 b}\\ &=-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{7/2} b}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{4 a^{7/2} b}\\ &=-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{8 a^{7/2} b^{3/2}}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,\sqrt{x}\right )}{8 a^{7/2} b^{3/2}}-\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,\sqrt{x}\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}\\ &=-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac{(b c-a d)^2 (11 b c+a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (11 b c+a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{\left ((b c-a d)^2 (11 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}\\ &=-\frac{c^2 (11 b c-7 a d)}{14 a^2 b x^{7/2}}+\frac{c \left (11 b^2 c^2-21 a b c d+6 a^2 d^2\right )}{6 a^3 b x^{3/2}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{7/2} \left (a+b x^2\right )}-\frac{(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (11 b c+a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{15/4} b^{5/4}}-\frac{(b c-a d)^2 (11 b c+a d) \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}+\frac{(b c-a d)^2 (11 b c+a d) \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{8 \sqrt{2} a^{15/4} b^{5/4}}\\ \end{align*}

Mathematica [C]  time = 1.799, size = 353, normalized size = 0.94 \[ -\frac{-229376 a b^2 x^4 \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{-\frac{3}{4},2,2,2,2\right \},\left \{1,1,1,\frac{13}{4}\right \},-\frac{b x^2}{a}\right )+315 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};-\frac{b x^2}{a}\right ) \left (3 a^2 b x^2 \left (3 c^2 d x^2+c^3-1149 c d^2 x^4+d^3 x^6\right )+a^3 \left (1875 c^2 d x^2+625 c^3+1875 c d^2 x^4+241 d^3 x^6\right )+9 a b^2 x^4 \left (977 c^2 d x^2+27 c^3+81 c d^2 x^4+27 d^3 x^6\right )+b^3 x^6 \left (7203 c^2 d x^2-1823 c^3+7203 c d^2 x^4+2401 d^3 x^6\right )\right )-15 a \left (21 a^2 \left (1875 c^2 d x^2+625 c^3+1875 c d^2 x^4+241 d^3 x^6\right )+6 a b x^2 \left (-6657 c^2 d x^2-1195 c^3+2751 c d^2 x^4+917 d^3 x^6\right )-7 b^2 x^4 \left (7203 c^2 d x^2-1823 c^3+7203 c d^2 x^4+2401 d^3 x^6\right )\right )}{241920 a^4 b x^{11/2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x^2)^3/(x^(9/2)*(a + b*x^2)^2),x]

[Out]

-(-15*a*(21*a^2*(625*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 241*d^3*x^6) + 6*a*b*x^2*(-1195*c^3 - 6657*c^2*d*
x^2 + 2751*c*d^2*x^4 + 917*d^3*x^6) - 7*b^2*x^4*(-1823*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6))
+ 315*(3*a^2*b*x^2*(c^3 + 3*c^2*d*x^2 - 1149*c*d^2*x^4 + d^3*x^6) + 9*a*b^2*x^4*(27*c^3 + 977*c^2*d*x^2 + 81*c
*d^2*x^4 + 27*d^3*x^6) + a^3*(625*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 241*d^3*x^6) + b^3*x^6*(-1823*c^3 +
7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 2401*d^3*x^6))*Hypergeometric2F1[1/4, 1, 5/4, -((b*x^2)/a)] - 229376*a*b^2*x
^4*(c + d*x^2)^3*HypergeometricPFQ[{-3/4, 2, 2, 2, 2}, {1, 1, 1, 13/4}, -((b*x^2)/a)])/(241920*a^4*b*x^(11/2))

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Maple [B]  time = 0.021, size = 706, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x)

[Out]

-2/7*c^3/a^2/x^(7/2)-2*c^2/a^2/x^(3/2)*d+4/3*c^3/a^3/x^(3/2)*b-1/2/b*x^(1/2)/(b*x^2+a)*d^3+3/2/a*x^(1/2)/(b*x^
2+a)*c*d^2-3/2/a^2*b*x^(1/2)/(b*x^2+a)*c^2*d+1/2/a^3*b^2*x^(1/2)/(b*x^2+a)*c^3+1/8/a/b*(1/b*a)^(1/4)*2^(1/2)*a
rctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*d^3+9/8/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-
1)*c*d^2-21/8/a^3*b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^2*d+11/8/a^4*b^2*(1/b*a)^(
1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^3+1/16/a/b*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^
(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*d^3+9/16/a^2*(1/b*a)^(1/4)*2^(1/
2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c*d^2-2
1/16/a^3*b*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2
^(1/2)+(1/b*a)^(1/2)))*c^2*d+11/16/a^4*b^2*(1/b*a)^(1/4)*2^(1/2)*ln((x+(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(
1/2))/(x-(1/b*a)^(1/4)*x^(1/2)*2^(1/2)+(1/b*a)^(1/2)))*c^3+1/8/a/b*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a
)^(1/4)*x^(1/2)+1)*d^3+9/8/a^2*(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c*d^2-21/8/a^3*b*
(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^2*d+11/8/a^4*b^2*(1/b*a)^(1/4)*2^(1/2)*arctan(
2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^3

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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((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.4073, size = 4648, normalized size = 12.36 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

1/168*(84*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676
588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*
c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/
(a^15*b^5))^(1/4)*arctan((sqrt(a^8*b^2*sqrt(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^
2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a
^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12
*d^12)/(a^15*b^5)) + (121*b^6*c^6 - 462*a*b^5*c^5*d + 639*a^2*b^4*c^4*d^2 - 356*a^3*b^3*c^3*d^3 + 39*a^4*b^2*c
^2*d^4 + 18*a^5*b*c*d^5 + a^6*d^6)*x)*a^11*b^4*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^1
0*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 56
88*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 +
a^12*d^12)/(a^15*b^5))^(3/4) - (11*a^11*b^7*c^3 - 21*a^12*b^6*c^2*d + 9*a^13*b^5*c*d^2 + a^14*b^4*d^3)*sqrt(x)
*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^
8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 6
92*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(3/4))/(14641*b^12*c^12
- 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*
a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9
+ 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)) + 21*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 1
11804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*
b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 40
2*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4)*log(a^4*b*(-(14641*b^12*c^12 - 111804*a*
b^11*c^11*d + 368082*a^2*b^10*c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*
d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b
^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12*d^12)/(a^15*b^5))^(1/4) + (11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 +
a^3*d^3)*sqrt(x)) - 21*(a^3*b^2*x^6 + a^4*b*x^4)*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*
c^10*d^2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 -
5688*a^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11
+ a^12*d^12)/(a^15*b^5))^(1/4)*log(-a^4*b*(-(14641*b^12*c^12 - 111804*a*b^11*c^11*d + 368082*a^2*b^10*c^10*d^
2 - 676588*a^3*b^9*c^9*d^3 + 746703*a^4*b^8*c^8*d^4 - 486648*a^5*b^7*c^7*d^5 + 160188*a^6*b^6*c^6*d^6 - 5688*a
^7*b^5*c^5*d^7 - 10017*a^8*b^4*c^4*d^8 + 692*a^9*b^3*c^3*d^9 + 402*a^10*b^2*c^2*d^10 + 36*a^11*b*c*d^11 + a^12
*d^12)/(a^15*b^5))^(1/4) + (11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*sqrt(x)) - 4*(12*a^2*b*c^3
- 7*(11*b^3*c^3 - 21*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 3*a^3*d^3)*x^4 - 4*(11*a*b^2*c^3 - 21*a^2*b*c^2*d)*x^2)*sqr
t(x))/(a^3*b^2*x^6 + a^4*b*x^4)

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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((d*x**2+c)**3/x**(9/2)/(b*x**2+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 1.19982, size = 687, normalized size = 1.83 \begin{align*} \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{4} b^{2}} + \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} - \frac{\sqrt{2}{\left (11 \, \left (a b^{3}\right )^{\frac{1}{4}} b^{3} c^{3} - 21 \, \left (a b^{3}\right )^{\frac{1}{4}} a b^{2} c^{2} d + 9 \, \left (a b^{3}\right )^{\frac{1}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{1}{4}} a^{3} d^{3}\right )} \log \left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, a^{4} b^{2}} + \frac{b^{3} c^{3} \sqrt{x} - 3 \, a b^{2} c^{2} d \sqrt{x} + 3 \, a^{2} b c d^{2} \sqrt{x} - a^{3} d^{3} \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{2 \,{\left (14 \, b c^{3} x^{2} - 21 \, a c^{2} d x^{2} - 3 \, a c^{3}\right )}}{21 \, a^{3} x^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)^3/x^(9/2)/(b*x^2+a)^2,x, algorithm="giac")

[Out]

1/8*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(
1/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^2) + 1/8*sqrt(2)*(11*(a
*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*arct
an(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^2) + 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*
c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d + 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*
(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 1/16*sqrt(2)*(11*(a*b^3)^(1/4)*b^3*c^3 - 21*(a*b^3)^(1/4)*a*b^2*c^2*d
+ 9*(a*b^3)^(1/4)*a^2*b*c*d^2 + (a*b^3)^(1/4)*a^3*d^3)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4
*b^2) + 1/2*(b^3*c^3*sqrt(x) - 3*a*b^2*c^2*d*sqrt(x) + 3*a^2*b*c*d^2*sqrt(x) - a^3*d^3*sqrt(x))/((b*x^2 + a)*a
^3*b) + 2/21*(14*b*c^3*x^2 - 21*a*c^2*d*x^2 - 3*a*c^3)/(a^3*x^(7/2))