### 3.675 $$\int \frac{(a+c x^2)^{5/2}}{(d+e x)^{9/2}} \, dx$$

Optimal. Leaf size=498 $-\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}$

[Out]

(8*c^2*(d*(32*c*d^2 + 29*a*e^2) + e*(8*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/(21*e^5*(c*d^2 + a*e^2)*Sqrt[d + e
*x]) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(11*c*d^2 + 5*a*e^2)*x)*(a + c*x^2)^(3/2))/(21*e^3*(c*d^2 + a*e^2)*(d +
e*x)^(5/2)) - (2*(a + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (16*Sqrt[-a]*c^(5/2)*d*(32*c*d^2 + 29*a*e^2)*Sqrt
[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqr
t[c]*d - a*e)])/(21*e^6*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) -
(16*Sqrt[-a]*c^(3/2)*(32*c*d^2 + 5*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/
a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*Sqr
t[d + e*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.454597, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {733, 811, 813, 844, 719, 424, 419} $\frac{8 c^2 \sqrt{a+c x^2} \left (e x \left (5 a e^2+8 c d^2\right )+d \left (29 a e^2+32 c d^2\right )\right )}{21 e^5 \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{16 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \left (5 a e^2+32 c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \sqrt{d+e x}}+\frac{16 \sqrt{-a} c^{5/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (29 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{4 c \left (a+c x^2\right )^{3/2} \left (e x \left (5 a e^2+11 c d^2\right )+2 d \left (a e^2+4 c d^2\right )\right )}{21 e^3 (d+e x)^{5/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(8*c^2*(d*(32*c*d^2 + 29*a*e^2) + e*(8*c*d^2 + 5*a*e^2)*x)*Sqrt[a + c*x^2])/(21*e^5*(c*d^2 + a*e^2)*Sqrt[d + e
*x]) - (4*c*(2*d*(4*c*d^2 + a*e^2) + e*(11*c*d^2 + 5*a*e^2)*x)*(a + c*x^2)^(3/2))/(21*e^3*(c*d^2 + a*e^2)*(d +
e*x)^(5/2)) - (2*(a + c*x^2)^(5/2))/(7*e*(d + e*x)^(7/2)) + (16*Sqrt[-a]*c^(5/2)*d*(32*c*d^2 + 29*a*e^2)*Sqrt
[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqr
t[c]*d - a*e)])/(21*e^6*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2]) -
(16*Sqrt[-a]*c^(3/2)*(32*c*d^2 + 5*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[1 + (c*x^2)/
a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(21*e^6*Sqr
t[d + e*x]*Sqrt[a + c*x^2])

Rule 733

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 1)), x] - Dist[(2*c*p)/(e*(m + 1)), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 811

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*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e
^2) + 2*c*d*p*(e*f - d*g))*x))/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)), x] - Dist[p/(e^2*(m + 1)*(m + 2)*(c*d^2
+ a*e^2)), Int[(d + e*x)^(m + 2)*(a + c*x^2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1
) - e*f*(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2
, 0] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] &&  !ILtQ[m + 2*p + 3, 0]

Rule 813

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*(a + c*x^2)^p)/(e^2*(m + 1)*(m + 2*p + 2)), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 719

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*a*Rt[-(c/a), 2]*(d + e*x)^m*Sqrt[
1 + (c*x^2)/a])/(c*Sqrt[a + c*x^2]*((c*(d + e*x))/(c*d - a*e*Rt[-(c/a), 2]))^m), Subst[Int[(1 + (2*a*e*Rt[-(c/
a), 2]*x^2)/(c*d - a*e*Rt[-(c/a), 2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-(c/a), 2]*x)/2]], x] /; FreeQ[{a,
c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{\left (a+c x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx &=-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{(10 c) \int \frac{x \left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx}{7 e}\\ &=-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{(4 c) \int \frac{\left (3 a c d e-c \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{(d+e x)^{3/2}} \, dx}{7 e^3 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{(8 c) \int \frac{a c e \left (8 c d^2+5 a e^2\right )-c^2 d \left (32 c d^2+29 a e^2\right ) x}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{21 e^5 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{\left (8 c^2 \left (32 c d^2+5 a e^2\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+c x^2}} \, dx}{21 e^6}-\frac{\left (8 c^3 d \left (32 c d^2+29 a e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+c x^2}} \, dx}{21 e^6 \left (c d^2+a e^2\right )}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}-\frac{\left (16 a c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{21 \sqrt{-a} e^6 \left (c d^2+a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{a+c x^2}}+\frac{\left (16 a c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt{\frac{c (d+e x)}{c d-\frac{a \sqrt{c} e}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 a \sqrt{c} e x^2}{\sqrt{-a} \left (c d-\frac{a \sqrt{c} e}{\sqrt{-a}}\right )}}} \, dx,x,\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )}{21 \sqrt{-a} e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ &=\frac{8 c^2 \left (d \left (32 c d^2+29 a e^2\right )+e \left (8 c d^2+5 a e^2\right ) x\right ) \sqrt{a+c x^2}}{21 e^5 \left (c d^2+a e^2\right ) \sqrt{d+e x}}-\frac{4 c \left (2 d \left (4 c d^2+a e^2\right )+e \left (11 c d^2+5 a e^2\right ) x\right ) \left (a+c x^2\right )^{3/2}}{21 e^3 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}-\frac{2 \left (a+c x^2\right )^{5/2}}{7 e (d+e x)^{7/2}}+\frac{16 \sqrt{-a} c^{5/2} d \left (32 c d^2+29 a e^2\right ) \sqrt{d+e x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \left (c d^2+a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{a+c x^2}}-\frac{16 \sqrt{-a} c^{3/2} \left (32 c d^2+5 a e^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{c} d+\sqrt{-a} e}} \sqrt{1+\frac{c x^2}{a}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{21 e^6 \sqrt{d+e x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 3.99691, size = 677, normalized size = 1.36 $\frac{\sqrt{d+e x} \left (\frac{2 \left (a+c x^2\right ) \left (\frac{2 c^2 d \left (67 a e^2+79 c d^2\right )}{(d+e x) \left (a e^2+c d^2\right )}-\frac{4 c \left (4 a e^2+13 c d^2\right )}{(d+e x)^2}+\frac{18 c d \left (a e^2+c d^2\right )}{(d+e x)^3}-\frac{3 \left (a e^2+c d^2\right )^2}{(d+e x)^4}+7 c^2\right )}{e^5}-\frac{16 c^2 \left (-\sqrt{a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+8 i \sqrt{a} c d^2 e+29 a \sqrt{c} d e^2+32 c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right ),\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+d e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (29 a^2 e^2+a c \left (32 d^2+29 e^2 x^2\right )+32 c^2 d^2 x^2\right )+\sqrt{c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3+32 \sqrt{a} c d^2 e-29 i a \sqrt{c} d e^2-32 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{e^7 (d+e x) \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (a e^2+c d^2\right )}\right )}{21 \sqrt{a+c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(9/2),x]

[Out]

(Sqrt[d + e*x]*((2*(a + c*x^2)*(7*c^2 - (3*(c*d^2 + a*e^2)^2)/(d + e*x)^4 + (18*c*d*(c*d^2 + a*e^2))/(d + e*x)
^3 - (4*c*(13*c*d^2 + 4*a*e^2))/(d + e*x)^2 + (2*c^2*d*(79*c*d^2 + 67*a*e^2))/((c*d^2 + a*e^2)*(d + e*x))))/e^
5 - (16*c^2*(d*e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(29*a^2*e^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 29*e^2*x^2))
+ Sqrt[c]*d*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (29*I)*a*Sqrt[c]*d*e^2 + 29*a^(3/2)*e^3)*Sqrt[(e*((I*S
qrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*
ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]
- Sqrt[a]*e*(32*c^(3/2)*d^3 + (8*I)*Sqrt[a]*c*d^2*e + 29*a*Sqrt[c]*d*e^2 + (5*I)*a^(3/2)*e^3)*Sqrt[(e*((I*Sqrt
[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*Arc
Sinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)]))/(
e^7*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(c*d^2 + a*e^2)*(d + e*x))))/(21*Sqrt[a + c*x^2])

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Maple [B]  time = 0.286, size = 5303, normalized size = 10.7 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^(5/2)/(e*x+d)^(9/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="fricas")

[Out]

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*sqrt(c*x^2 + a)*sqrt(e*x + d)/(e^5*x^5 + 5*d*e^4*x^4 + 10*d^2*e^3*x^3 + 1
0*d^3*e^2*x^2 + 5*d^4*e*x + d^5), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + c x^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{\frac{9}{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**(5/2)/(e*x+d)**(9/2),x)

[Out]

Integral((a + c*x**2)**(5/2)/(d + e*x)**(9/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + a\right )}^{\frac{5}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^(5/2)/(e*x+d)^(9/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^(5/2)/(e*x + d)^(9/2), x)