3.443 \(\int \frac{\sqrt{x} (c+d x^2)^3}{a+b x^2} \, dx\)
Optimal. Leaf size=306 \[ \frac{2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{2 d^3 x^{11/2}}{11 b} \]
[Out]
(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11
/2))/(11*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c
- a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4))
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Rubi [A] time = 0.245434, antiderivative size = 306, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used
= {461, 329, 297, 1162, 617, 204, 1165, 628} \[ \frac{2 d x^{3/2} \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{3 b^3}+\frac{2 d^2 x^{7/2} (3 b c-a d)}{7 b^2}+\frac{(b c-a d)^3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{2 d^3 x^{11/2}}{11 b} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
(2*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x^(3/2))/(3*b^3) + (2*d^2*(3*b*c - a*d)*x^(7/2))/(7*b^2) + (2*d^3*x^(11
/2))/(11*b) - ((b*c - a*d)^3*ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c
- a*d)^3*ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)])/(Sqrt[2]*a^(1/4)*b^(15/4)) + ((b*c - a*d)^3*Log[Sqrt[
a] - Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4)) - ((b*c - a*d)^3*Log[Sqrt[a] +
Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x])/(2*Sqrt[2]*a^(1/4)*b^(15/4))
Rule 461
Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] || !RationalQ[m])
Rule 329
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
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{\sqrt{x} \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sqrt{x}}{b^3}+\frac{d^2 (3 b c-a d) x^{5/2}}{b^2}+\frac{d^3 x^{9/2}}{b}+\frac{\left (b^3 c^3-3 a b^2 c^2 d+3 a^2 b c d^2-a^3 d^3\right ) \sqrt{x}}{b^3 \left (a+b x^2\right )}\right ) \, dx\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}+\frac{(b c-a d)^3 \int \frac{\sqrt{x}}{a+b x^2} \, dx}{b^3}\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}+\frac{\left (2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^3}\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^{7/2}}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,\sqrt{x}\right )}{b^{7/2}}\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}+\frac{(b c-a d)^3 \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 )}{2 b^4}+\frac{(b c-a d)^3 \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 )}{2 b^4}+\frac{(b c-a d)^3 \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 )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \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 )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}+\frac{(b c-a d)^3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}\\ &=\frac{2 d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^{3/2}}{3 b^3}+\frac{2 d^2 (3 b c-a d) x^{7/2}}{7 b^2}+\frac{2 d^3 x^{11/2}}{11 b}-\frac{(b c-a d)^3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt{2} \sqrt [4]{a} b^{15/4}}+\frac{(b c-a d)^3 \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}-\frac{(b c-a d)^3 \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{b} x\right )}{2 \sqrt{2} \sqrt [4]{a} b^{15/4}}\\ \end{align*}
Mathematica [C] time = 0.359545, size = 97, normalized size = 0.32 \[ \frac{2 x^{3/2} \left (a d \left (77 a^2 d^2-33 a b d \left (7 c+d x^2\right )+3 b^2 \left (77 c^2+33 c d x^2+7 d^2 x^4\right )\right )+77 (b c-a d)^3 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{b x^2}{a}\right )\right )}{231 a b^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[x]*(c + d*x^2)^3)/(a + b*x^2),x]
[Out]
(2*x^(3/2)*(a*d*(77*a^2*d^2 - 33*a*b*d*(7*c + d*x^2) + 3*b^2*(77*c^2 + 33*c*d*x^2 + 7*d^2*x^4)) + 77*(b*c - a*
d)^3*Hypergeometric2F1[3/4, 1, 7/4, -((b*x^2)/a)]))/(231*a*b^3)
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Maple [B] time = 0.01, size = 659, normalized size = 2.2 \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^(1/2)/(b*x^2+a),x)
[Out]
2/11*d^3*x^(11/2)/b-2/7*d^3/b^2*x^(7/2)*a+6/7*d^2/b*x^(7/2)*c+2/3*d^3/b^3*x^(3/2)*a^2-2*d^2/b^2*x^(3/2)*c*a+2*
d/b*x^(3/2)*c^2-1/2/b^4/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*a^3*d^3+3/2/b^3/(1/b*a)^
(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*a^2*c*d^2-3/2/b^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
1/b*a)^(1/4)*x^(1/2)+1)*a*c^2*d+1/2/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)+1)*c^3-1/2/b^
4/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*a^3*d^3+3/2/b^3/(1/b*a)^(1/4)*2^(1/2)*arctan(2
^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*a^2*c*d^2-3/2/b^2/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1
)*a*c^2*d+1/2/b/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^(1/2)-1)*c^3-1/4/b^4/(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)))*a^3*d^3+
3/4/b^3/(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)))*a^2*c*d^2-3/4/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)))*a*c^2*d+1/4/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
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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^(1/2)/(b*x^2+a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.6619, size = 5393, normalized size = 17.62 \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^(1/2)/(b*x^2+a),x, algorithm="fricas")
[Out]
1/462*(924*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*
d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*
d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*arctan((sqrt((b^18*c^18 - 18*a*b^17
*c^17*d + 153*a^2*b^16*c^16*d^2 - 816*a^3*b^15*c^15*d^3 + 3060*a^4*b^14*c^14*d^4 - 8568*a^5*b^13*c^13*d^5 + 18
564*a^6*b^12*c^12*d^6 - 31824*a^7*b^11*c^11*d^7 + 43758*a^8*b^10*c^10*d^8 - 48620*a^9*b^9*c^9*d^9 + 43758*a^10
*b^8*c^8*d^10 - 31824*a^11*b^7*c^7*d^11 + 18564*a^12*b^6*c^6*d^12 - 8568*a^13*b^5*c^5*d^13 + 3060*a^14*b^4*c^4
*d^14 - 816*a^15*b^3*c^3*d^15 + 153*a^16*b^2*c^2*d^16 - 18*a^17*b*c*d^17 + a^18*d^18)*x - (a*b^19*c^12 - 12*a^
2*b^18*c^11*d + 66*a^3*b^17*c^10*d^2 - 220*a^4*b^16*c^9*d^3 + 495*a^5*b^15*c^8*d^4 - 792*a^6*b^14*c^7*d^5 + 92
4*a^7*b^13*c^6*d^6 - 792*a^8*b^12*c^5*d^7 + 495*a^9*b^11*c^4*d^8 - 220*a^10*b^10*c^3*d^9 + 66*a^11*b^9*c^2*d^1
0 - 12*a^12*b^8*c*d^11 + a^13*b^7*d^12)*sqrt(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b
^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b
^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15)))*b^4*(-(b^12
*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*
d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*
d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4) + (b^13*c^9 - 9*a*b^12*c^8*d + 36*a^2*b^11*c^7*d^2 - 84*a
^3*b^10*c^6*d^3 + 126*a^4*b^9*c^5*d^4 - 126*a^5*b^8*c^4*d^5 + 84*a^6*b^7*c^3*d^6 - 36*a^7*b^6*c^2*d^7 + 9*a^8*
b^5*c*d^8 - a^9*b^4*d^9)*sqrt(x)*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3
+ 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8
- 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4))/(b^12*c^12 - 12*
a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a
^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a
^11*b*c*d^11 + a^12*d^12)) - 231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*
d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*
d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(a*b^11*(-
(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7
*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2
*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 - 84
*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*a^8
*b*c*d^8 - a^9*d^9)*sqrt(x)) + 231*b^3*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^
9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^
4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(1/4)*log(-a*b^11
*(-(b^12*c^12 - 12*a*b^11*c^11*d + 66*a^2*b^10*c^10*d^2 - 220*a^3*b^9*c^9*d^3 + 495*a^4*b^8*c^8*d^4 - 792*a^5*
b^7*c^7*d^5 + 924*a^6*b^6*c^6*d^6 - 792*a^7*b^5*c^5*d^7 + 495*a^8*b^4*c^4*d^8 - 220*a^9*b^3*c^3*d^9 + 66*a^10*
b^2*c^2*d^10 - 12*a^11*b*c*d^11 + a^12*d^12)/(a*b^15))^(3/4) - (b^9*c^9 - 9*a*b^8*c^8*d + 36*a^2*b^7*c^7*d^2 -
84*a^3*b^6*c^6*d^3 + 126*a^4*b^5*c^5*d^4 - 126*a^5*b^4*c^4*d^5 + 84*a^6*b^3*c^3*d^6 - 36*a^7*b^2*c^2*d^7 + 9*
a^8*b*c*d^8 - a^9*d^9)*sqrt(x)) + 4*(21*b^2*d^3*x^5 + 33*(3*b^2*c*d^2 - a*b*d^3)*x^3 + 77*(3*b^2*c^2*d - 3*a*b
*c*d^2 + a^2*d^3)*x)*sqrt(x))/b^3
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x**2+c)**3*x**(1/2)/(b*x**2+a),x)
[Out]
Exception raised: TypeError
________________________________________________________________________________________
Giac [B] time = 1.19675, size = 662, normalized size = 2.16 \begin{align*} \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a b^{6}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{2 \, a b^{6}} - \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a b^{6}} + \frac{\sqrt{2}{\left (\left (a b^{3}\right )^{\frac{3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac{3}{4}} a b^{2} c^{2} d + 3 \, \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 )}{4 \, a b^{6}} + \frac{2 \,{\left (21 \, b^{10} d^{3} x^{\frac{11}{2}} + 99 \, b^{10} c d^{2} x^{\frac{7}{2}} - 33 \, a b^{9} d^{3} x^{\frac{7}{2}} + 231 \, b^{10} c^{2} d x^{\frac{3}{2}} - 231 \, a b^{9} c d^{2} x^{\frac{3}{2}} + 77 \, a^{2} b^{8} d^{3} x^{\frac{3}{2}}\right )}}{231 \, b^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x^2+c)^3*x^(1/2)/(b*x^2+a),x, algorithm="giac")
[Out]
1/2*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(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*b^6) + 1/2*sqrt(2)*((a*b^3)^(3/
4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(a*b^3)^(3/4)*a^2*b*c*d^2 - (a*b^3)^(3/4)*a^3*d^3)*arctan(-1/2*sq
rt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a*b^6) - 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^
(3/4)*a*b^2*c^2*d + 3*(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*b^6) + 1/4*sqrt(2)*((a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d + 3*(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*b^6) + 2/231*(21*b^10*d^3
*x^(11/2) + 99*b^10*c*d^2*x^(7/2) - 33*a*b^9*d^3*x^(7/2) + 231*b^10*c^2*d*x^(3/2) - 231*a*b^9*c*d^2*x^(3/2) +
77*a^2*b^8*d^3*x^(3/2))/b^11