3.459 \(\int \frac{(c+d x^2)^3}{x^{7/2} (a+b x^2)^2} \, dx\)

Optimal. Leaf size=376 \[ \frac{c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt{x}}+\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \]

[Out]

-(c^2*(9*b*c - 5*a*d))/(10*a^2*b*x^(5/2)) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d^2))/(2*a^3*b*Sqrt[x]) + ((b*c
- a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4))

________________________________________________________________________________________

Rubi [A]  time = 0.429371, 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, 297, 1162, 617, 204, 1165, 628} \[ \frac{c \left (2 a^2 d^2-15 a b c d+9 b^2 c^2\right )}{2 a^3 b \sqrt{x}}+\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} a^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (a d+3 b c) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^{5/2} \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c^2*(9*b*c - 5*a*d))/(10*a^2*b*x^(5/2)) + (c*(9*b^2*c^2 - 15*a*b*c*d + 2*a^2*d^2))/(2*a^3*b*Sqrt[x]) + ((b*c
- a*d)*(c + d*x^2)^2)/(2*a*b*x^(5/2)*(a + b*x^2)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4
)*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)
*Sqrt[x])/a^(1/4)])/(4*Sqrt[2]*a^(13/4)*b^(7/4)) + (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4
)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4)) - (3*(b*c - a*d)^2*(3*b*c + a*d)*Log[Sqrt[a] + Sq
rt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(8*Sqrt[2]*a^(13/4)*b^(7/4))

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 297

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

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{\left (c+d x^2\right )^3}{x^{7/2} \left (a+b x^2\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (c+d x^4\right )^3}{x^6 \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^{5/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \frac{\left (c+d x^4\right ) \left (-c (9 b c-5 a d)-d (b c+3 a d) x^4\right )}{x^6 \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^{5/2} \left (a+b x^2\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{c^2 (-9 b c+5 a d)}{a x^6}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{a^2 x^2}-\frac{3 (-b c+a d)^2 (3 b c+a d) x^2}{a^2 \left (a+b x^4\right )}\right ) \, dx,x,\sqrt{x}\right )}{2 a b}\\ &=-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt{x}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac{\left (3 (b c-a d)^2 (3 b c+a d)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{2 a^3 b}\\ &=-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt{x}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac{\left (3 (b c-a d)^2 (3 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^3 b^{3/2}}+\frac{\left (3 (b c-a d)^2 (3 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^3 b^{3/2}}\\ &=-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt{x}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac{\left (3 (b c-a d)^2 (3 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^3 b^2}+\frac{\left (3 (b c-a d)^2 (3 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^3 b^2}+\frac{\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}+\frac{\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ &=-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt{x}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}+\frac{3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}+\frac{\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac{\left (3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ &=-\frac{c^2 (9 b c-5 a d)}{10 a^2 b x^{5/2}}+\frac{c \left (9 b^2 c^2-15 a b c d+2 a^2 d^2\right )}{2 a^3 b \sqrt{x}}+\frac{(b c-a d) \left (c+d x^2\right )^2}{2 a b x^{5/2} \left (a+b x^2\right )}-\frac{3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (3 b c+a d) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{13/4} b^{7/4}}+\frac{3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}-\frac{3 (b c-a d)^2 (3 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^{13/4} b^{7/4}}\\ \end{align*}

Mathematica [C]  time = 1.89733, size = 353, normalized size = 0.94 \[ -\frac{-491520 a b^2 x^4 \left (c+d x^2\right )^3 \text{HypergeometricPFQ}\left (\left \{-\frac{1}{4},2,2,2,2\right \},\left \{1,1,1,\frac{15}{4}\right \},-\frac{b x^2}{a}\right )+385 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right ) \left (9 a^2 b x^2 \left (81 c^2 d x^2+27 c^3-47 c d^2 x^4+27 d^3 x^6\right )+a^3 \left (7203 c^2 d x^2+2401 c^3+7203 c d^2 x^4+1249 d^3 x^6\right )+3 a b^2 x^4 \left (1923 c^2 d x^2+c^3+3 c d^2 x^4+d^3 x^6\right )+b^3 x^6 \left (1875 c^2 d x^2-2831 c^3+1875 c d^2 x^4+625 d^3 x^6\right )\right )-77 a \left (5 a^2 \left (7203 c^2 d x^2+2401 c^3+7203 c d^2 x^4+1249 d^3 x^6\right )+6 a b x^2 \left (-13485 c^2 d x^2-1423 c^3+915 c d^2 x^4+305 d^3 x^6\right )-15 b^2 x^4 \left (1875 c^2 d x^2-2831 c^3+1875 c d^2 x^4+625 d^3 x^6\right )\right )}{887040 a^4 b x^{9/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-77*a*(6*a*b*x^2*(-1423*c^3 - 13485*c^2*d*x^2 + 915*c*d^2*x^4 + 305*d^3*x^6) - 15*b^2*x^4*(-2831*c^3 + 1875*
c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + 5*a^2*(2401*c^3 + 7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 1249*d^3*x^6))
+ 385*(3*a*b^2*x^4*(c^3 + 1923*c^2*d*x^2 + 3*c*d^2*x^4 + d^3*x^6) + 9*a^2*b*x^2*(27*c^3 + 81*c^2*d*x^2 - 47*c
*d^2*x^4 + 27*d^3*x^6) + b^3*x^6*(-2831*c^3 + 1875*c^2*d*x^2 + 1875*c*d^2*x^4 + 625*d^3*x^6) + a^3*(2401*c^3 +
7203*c^2*d*x^2 + 7203*c*d^2*x^4 + 1249*d^3*x^6))*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)] - 491520*a*b^2*
x^4*(c + d*x^2)^3*HypergeometricPFQ[{-1/4, 2, 2, 2, 2}, {1, 1, 1, 15/4}, -((b*x^2)/a)])/(887040*a^4*b*x^(9/2))

________________________________________________________________________________________

Maple [B]  time = 0.02, size = 697, 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^(7/2)/(b*x^2+a)^2,x)

[Out]

-2/5*c^3/a^2/x^(5/2)-6*c^2/a^2/x^(1/2)*d+4*c^3/a^3/x^(1/2)*b-1/2/b*x^(3/2)/(b*x^2+a)*d^3+3/2/a*x^(3/2)/(b*x^2+
a)*c*d^2-3/2/a^2*b*x^(3/2)/(b*x^2+a)*c^2*d+1/2/a^3*b^2*x^(3/2)/(b*x^2+a)*c^3+3/8/b^2/(1/b*a)^(1/4)*2^(1/2)*arc
tan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*d^3+3/8/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)
*c*d^2-15/8/a^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^2*d+9/8/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^3+3/16/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)))*d^3+3/16/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)))*c*d^2-15/16/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^2*d+9/16/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^3+3/8/b^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1
/2)+1)*d^3+3/8/a/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c*d^2-15/8/a^2/(1/b*a)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^2*d+9/8/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^3

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.65603, size = 5704, normalized size = 15.17 \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^(7/2)/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/40*(60*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^
9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b
^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*arctan
((sqrt((729*b^18*c^18 - 7290*a*b^17*c^17*d + 31833*a^2*b^16*c^16*d^2 - 78192*a^3*b^15*c^15*d^3 + 113940*a^4*b^
14*c^14*d^4 - 88920*a^5*b^13*c^13*d^5 + 10180*a^6*b^12*c^12*d^6 + 46320*a^7*b^11*c^11*d^7 - 35970*a^8*b^10*c^1
0*d^8 - 220*a^9*b^9*c^9*d^9 + 12078*a^10*b^8*c^8*d^10 - 3600*a^11*b^7*c^7*d^11 - 1884*a^12*b^6*c^6*d^12 + 936*
a^13*b^5*c^5*d^13 + 180*a^14*b^4*c^4*d^14 - 112*a^15*b^3*c^3*d^15 - 15*a^16*b^2*c^2*d^16 + 6*a^17*b*c*d^17 + a
^18*d^18)*x - (81*a^7*b^15*c^12 - 540*a^8*b^14*c^11*d + 1458*a^9*b^13*c^10*d^2 - 1932*a^10*b^12*c^9*d^3 + 1039
*a^11*b^11*c^8*d^4 + 328*a^12*b^10*c^7*d^5 - 644*a^13*b^9*c^6*d^6 + 136*a^14*b^8*c^5*d^7 + 127*a^15*b^7*c^4*d^
8 - 44*a^16*b^6*c^3*d^9 - 14*a^17*b^5*c^2*d^10 + 4*a^18*b^4*c*d^11 + a^19*b^3*d^12)*sqrt(-(81*b^12*c^12 - 540*
a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 6
44*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4
*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7)))*a^3*b^2*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2
- 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^
7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))
^(1/4) - (27*a^3*b^11*c^9 - 135*a^4*b^10*c^8*d + 252*a^5*b^9*c^7*d^2 - 188*a^6*b^8*c^6*d^3 - 6*a^7*b^7*c^5*d^4
+ 78*a^8*b^6*c^4*d^5 - 20*a^9*b^5*c^3*d^6 - 12*a^10*b^4*c^2*d^7 + 3*a^11*b^3*c*d^8 + a^12*b^2*d^9)*sqrt(x)*(-
(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328
*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a
^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4))/(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^
2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 13
6*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d
^12)) - 15*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b
^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*
b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*log(2
7*a^10*b^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*
c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c
^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(3/4) + 27*(27*b^9*c^9 - 135*a*b^8*c^
8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6
- 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x)) + 15*(a^3*b^2*x^5 + a^4*b*x^3)*(-(81*b^12*c^12 - 540*
a*b^11*c^11*d + 1458*a^2*b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 6
44*a^6*b^6*c^6*d^6 + 136*a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4
*a^11*b*c*d^11 + a^12*d^12)/(a^13*b^7))^(1/4)*log(-27*a^10*b^5*(-(81*b^12*c^12 - 540*a*b^11*c^11*d + 1458*a^2*
b^10*c^10*d^2 - 1932*a^3*b^9*c^9*d^3 + 1039*a^4*b^8*c^8*d^4 + 328*a^5*b^7*c^7*d^5 - 644*a^6*b^6*c^6*d^6 + 136*
a^7*b^5*c^5*d^7 + 127*a^8*b^4*c^4*d^8 - 44*a^9*b^3*c^3*d^9 - 14*a^10*b^2*c^2*d^10 + 4*a^11*b*c*d^11 + a^12*d^1
2)/(a^13*b^7))^(3/4) + 27*(27*b^9*c^9 - 135*a*b^8*c^8*d + 252*a^2*b^7*c^7*d^2 - 188*a^3*b^6*c^6*d^3 - 6*a^4*b^
5*c^5*d^4 + 78*a^5*b^4*c^4*d^5 - 20*a^6*b^3*c^3*d^6 - 12*a^7*b^2*c^2*d^7 + 3*a^8*b*c*d^8 + a^9*d^9)*sqrt(x)) +
4*(4*a^2*b*c^3 - 5*(9*b^3*c^3 - 15*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^4 - 12*(3*a*b^2*c^3 - 5*a^2*b*c^2
*d)*x^2)*sqrt(x))/(a^3*b^2*x^5 + a^4*b*x^3)

________________________________________________________________________________________

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**(7/2)/(b*x**2+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.25349, size = 682, normalized size = 1.81 \begin{align*} \frac{b^{3} c^{3} x^{\frac{3}{2}} - 3 \, a b^{2} c^{2} d x^{\frac{3}{2}} + 3 \, a^{2} b c d^{2} x^{\frac{3}{2}} - a^{3} d^{3} x^{\frac{3}{2}}}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{2 \,{\left (10 \, b c^{3} x^{2} - 15 \, a c^{2} d x^{2} - a c^{3}\right )}}{5 \, a^{3} x^{\frac{5}{2}}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{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^{4}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{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^{4}} - \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{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^{4}} + \frac{3 \, \sqrt{2}{\left (3 \, \left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 5 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + \left (a b^{3}\right )^{\frac{3}{4}} a^{2} b c d^{2} + \left (a b^{3}\right )^{\frac{3}{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^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b^3*c^3*x^(3/2) - 3*a*b^2*c^2*d*x^(3/2) + 3*a^2*b*c*d^2*x^(3/2) - a^3*d^3*x^(3/2))/((b*x^2 + a)*a^3*b) +
2/5*(10*b*c^3*x^2 - 15*a*c^2*d*x^2 - a*c^3)/(a^3*x^(5/2)) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(
3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*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^4) + 3/8*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*
b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*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^4) - 3/16*sqrt(2)*(3*(a*b^3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d
^2 + (a*b^3)^(3/4)*a^3*d^3)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^4) + 3/16*sqrt(2)*(3*(a*b^
3)^(3/4)*b^3*c^3 - 5*(a*b^3)^(3/4)*a*b^2*c^2*d + (a*b^3)^(3/4)*a^2*b*c*d^2 + (a*b^3)^(3/4)*a^3*d^3)*log(-sqrt(
2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^4)