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

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

[Out]

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

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Rubi [A]  time = 0.45, antiderivative size = 498, normalized size of antiderivative = 1.00, 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 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)])

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 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 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]

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*}

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Mathematica [C]  time = 3.28, size = 677, normalized size = 1.36 \[ \frac {\sqrt {d+e x} \left (\frac {2 \left (a+c x^2\right ) \left (\frac {2 c^2 d \left (67 a e^2+79 c d^2\right )}{(d+e x) \left (a e^2+c d^2\right )}-\frac {4 c \left (4 a e^2+13 c d^2\right )}{(d+e x)^2}+\frac {18 c d \left (a e^2+c d^2\right )}{(d+e x)^3}-\frac {3 \left (a e^2+c d^2\right )^2}{(d+e x)^4}+7 c^2\right )}{e^5}-\frac {16 c^2 \left (-\sqrt {a} e (d+e x)^{3/2} \left (5 i a^{3/2} e^3+8 i \sqrt {a} c d^2 e+29 a \sqrt {c} d e^2+32 c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+\sqrt {c} d (d+e x)^{3/2} \left (29 a^{3/2} e^3+32 \sqrt {a} c d^2 e-29 i a \sqrt {c} d e^2-32 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+d e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (29 a^2 e^2+a c \left (32 d^2+29 e^2 x^2\right )+32 c^2 d^2 x^2\right )\right )}{e^7 (d+e x) \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (a e^2+c d^2\right )}\right )}{21 \sqrt {a+c x^2}} \]

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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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{5} x^{5} + 5 \, d e^{4} x^{4} + 10 \, d^{2} e^{3} x^{3} + 10 \, d^{3} e^{2} x^{2} + 5 \, d^{4} e x + d^{5}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]

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)

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maple [B]  time = 0.12, size = 5303, normalized size = 10.65 \[ \text {output too large to display} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {9}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{9/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {9}{2}}}\, dx \]

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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