3.674 \(\int \frac {(a+c x^2)^{5/2}}{(d+e x)^{7/2}} \, dx\)
Optimal. Leaf size=420 \[ \frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \left (17 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {16 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (9 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{15 e^5 \sqrt {d+e x}}+\frac {4 c \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}} \]
[Out]
4/15*c*(3*e*x+8*d)*(c*x^2+a)^(3/2)/e^3/(e*x+d)^(3/2)-2/5*(c*x^2+a)^(5/2)/e/(e*x+d)^(5/2)-8/15*c*(8*c*d*e*x+9*a
*e^2+32*c*d^2)*(c*x^2+a)^(1/2)/e^5/(e*x+d)^(1/2)-16/15*c^(3/2)*(9*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/(c*x^2+a)^(1/2)/((e*x+d)*c^(1/2)/(e*(-a)^(1/2)+d*c^(1/2)))^(1/2)+16/15*c^(3/2)*d*(17*a*e^2+32*c*d^2)*Ell
ipticF(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 = 0.36, antiderivative size = 420, normalized size of antiderivative = 1.00,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used =
{733, 813, 844, 719, 424, 419} \[ \frac {16 \sqrt {-a} c^{3/2} d \sqrt {\frac {c x^2}{a}+1} \left (17 a e^2+32 c d^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {a+c x^2} \sqrt {d+e x}}-\frac {16 \sqrt {-a} c^{3/2} \sqrt {\frac {c x^2}{a}+1} \sqrt {d+e x} \left (9 a e^2+32 c d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a e}{\sqrt {-a} \sqrt {c} d-a e}\right )}{15 e^6 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {-a} e+\sqrt {c} d}}}-\frac {8 c \sqrt {a+c x^2} \left (9 a e^2+32 c d^2+8 c d e x\right )}{15 e^5 \sqrt {d+e x}}+\frac {4 c \left (a+c x^2\right )^{3/2} (8 d+3 e x)}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + c*x^2)^(5/2)/(d + e*x)^(7/2),x]
[Out]
(-8*c*(32*c*d^2 + 9*a*e^2 + 8*c*d*e*x)*Sqrt[a + c*x^2])/(15*e^5*Sqrt[d + e*x]) + (4*c*(8*d + 3*e*x)*(a + c*x^2
)^(3/2))/(15*e^3*(d + e*x)^(3/2)) - (2*(a + c*x^2)^(5/2))/(5*e*(d + e*x)^(5/2)) - (16*Sqrt[-a]*c^(3/2)*(32*c*d
^2 + 9*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]*Sqrt[c]*d - a*e)])/(15*e^6*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x^2])
+ (16*Sqrt[-a]*c^(3/2)*d*(32*c*d^2 + 17*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)])/(15*e^
6*Sqrt[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 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)^{7/2}} \, dx &=-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {(2 c) \int \frac {x \left (a+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx}{e}\\ &=\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {(4 c) \int \frac {(-3 a e+8 c d x) \sqrt {a+c x^2}}{(d+e x)^{3/2}} \, dx}{5 e^3}\\ &=-\frac {8 c \left (32 c d^2+9 a e^2+8 c d e x\right ) \sqrt {a+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {(8 c) \int \frac {-8 a c d e+c \left (32 c d^2+9 a e^2\right ) x}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^5}\\ &=-\frac {8 c \left (32 c d^2+9 a e^2+8 c d e x\right ) \sqrt {a+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {\left (8 c^2 \left (32 c d^2+9 a e^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx}{15 e^6}-\frac {\left (8 c^2 d \left (32 c d^2+17 a e^2\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx}{15 e^6}\\ &=-\frac {8 c \left (32 c d^2+9 a e^2+8 c d e x\right ) \sqrt {a+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}+\frac {\left (16 a c^{3/2} \left (32 c d^2+9 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 )}{15 \sqrt {-a} e^6 \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} d \left (32 c d^2+17 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 )}{15 \sqrt {-a} e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ &=-\frac {8 c \left (32 c d^2+9 a e^2+8 c d e x\right ) \sqrt {a+c x^2}}{15 e^5 \sqrt {d+e x}}+\frac {4 c (8 d+3 e x) \left (a+c x^2\right )^{3/2}}{15 e^3 (d+e x)^{3/2}}-\frac {2 \left (a+c x^2\right )^{5/2}}{5 e (d+e x)^{5/2}}-\frac {16 \sqrt {-a} c^{3/2} \left (32 c d^2+9 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 )}{15 e^6 \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} d \left (32 c d^2+17 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 )}{15 e^6 \sqrt {d+e x} \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [C] time = 3.33, size = 613, normalized size = 1.46 \[ \frac {2 \left (-e^2 \left (a+c x^2\right ) \left (3 a^2 e^4+2 a c e^2 \left (10 d^2+25 d e x+18 e^2 x^2\right )+c^2 \left (128 d^4+288 d^3 e x+176 d^2 e^2 x^2+10 d e^3 x^3-3 e^4 x^4\right )\right )+\frac {8 c (d+e x)^2 \left (\sqrt {c} (d+e x)^{3/2} \left (9 a^{3/2} e^3+32 \sqrt {a} c d^2 e-9 i a \sqrt {c} d e^2-32 i c^{3/2} d^3\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )+e^2 \sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}} \left (9 a^2 e^2+a c \left (32 d^2+9 e^2 x^2\right )+32 c^2 d^2 x^2\right )-\sqrt {a} \sqrt {c} e (d+e x)^{3/2} \left (8 i \sqrt {a} \sqrt {c} d e+9 a e^2+32 c d^2\right ) \sqrt {\frac {e \left (x+\frac {i \sqrt {a}}{\sqrt {c}}\right )}{d+e x}} \sqrt {-\frac {-e x+\frac {i \sqrt {a} e}{\sqrt {c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac {\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d-i \sqrt {a} e}{\sqrt {c} d+i \sqrt {a} e}\right )\right )}{\sqrt {-d-\frac {i \sqrt {a} e}{\sqrt {c}}}}\right )}{15 e^7 \sqrt {a+c x^2} (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + c*x^2)^(5/2)/(d + e*x)^(7/2),x]
[Out]
(2*(-(e^2*(a + c*x^2)*(3*a^2*e^4 + 2*a*c*e^2*(10*d^2 + 25*d*e*x + 18*e^2*x^2) + c^2*(128*d^4 + 288*d^3*e*x + 1
76*d^2*e^2*x^2 + 10*d*e^3*x^3 - 3*e^4*x^4))) + (8*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*(9*a^2*e
^2 + 32*c^2*d^2*x^2 + a*c*(32*d^2 + 9*e^2*x^2)) + Sqrt[c]*((-32*I)*c^(3/2)*d^3 + 32*Sqrt[a]*c*d^2*e - (9*I)*a*
Sqrt[c]*d*e^2 + 9*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)*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]*Sqrt[c]*e*(32*c*d^2 + (8*I)*Sqrt[a]*Sqrt[c]*d*e + 9*a*e^
2)*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*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[-d - (I*Sqrt[a]*e)/Sqrt[c]]))/(15*e^7*(d + e*x)^(5/2)*Sqrt[a + c*x^2])
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt {c x^{2} + a} \sqrt {e x + d}}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^(7/2),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^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*
d^3*e*x + d^4), x)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="giac")
[Out]
integrate((c*x^2 + a)^(5/2)/(e*x + d)^(7/2), x)
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maple [B] time = 0.12, size = 3421, normalized size = 8.15 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c*x^2+a)^(5/2)/(e*x+d)^(7/2),x)
[Out]
2/15*(-60*a*c^2*d*e^5*x^3-196*a*c^2*d^2*e^4*x^2-288*a*c^2*d^3*e^3*x-10*c^3*d*e^5*x^5-288*c^3*d^3*e^3*x^3-39*a^
2*c*e^6*x^2-33*a*c^2*e^6*x^4-176*c^3*d^2*e^4*x^4+3*c^3*e^6*x^6-20*a^2*c*d^2*e^4-128*a*c^2*d^4*e^2-256*Elliptic
E((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^3*d^6*(-(e*x
+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*
d+(-a*c)^(1/2)*e)*e)^(1/2)-3*a^3*e^6-128*c^3*d^4*e^2*x^2+192*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2
),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c^2*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1
/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+256*
EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*c^
2*d^3*e^3*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(
1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-328*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)
,(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c^2*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/
2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+144*E
llipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a^2*c
*d*e^5*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c
)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+384*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^
(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a*c^2*d^3*e^3*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(
1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+512*EllipticF((-(e*x+d)
/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*c^2*d^4*e^2*(-a*c)^(1/2
)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/
2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-144*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)
*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a^2*c*d*e^5*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(
c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-656*EllipticE((-(e*x+d)/(-c*d+
(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*a*c^2*d^3*e^3*(-(e*x+d)/(-c*d+
(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(
1/2)*e)*e)^(1/2)+136*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1
/2)*e))^(1/2))*a*c*d^3*e^3*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a
*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+136*EllipticF((-(e*x+d)/(-c*d+(-a*c)^
(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a*c*d*e^5*(-a*c)^(1/2)*(-(e*x+d)/(-
c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*
c)^(1/2)*e)*e)^(1/2)+272*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c
)^(1/2)*e))^(1/2))*x*a*c*d^2*e^4*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c
*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-256*EllipticE((-(e*x+d)/(-c*d+(
-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*c^3*d^4*e^2*(-(e*x+d)/(-c*d+(
-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1
/2)*e)*e)^(1/2)-328*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/
2)*e))^(1/2))*a*c^2*d^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)
*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+72*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(
1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a^2*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/
2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+192*E
llipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*a*c^2*d
^4*e^2*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c
)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-512*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^
(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x*c^3*d^5*e*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2)
)/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+72*EllipticF((-(e*x+d)/(-c*
d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*x^2*a^2*c*e^6*(-(e*x+d)/(-c*d+
(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(
1/2)*e)*e)^(1/2)-72*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/
2)*e))^(1/2))*x^2*a^2*c*e^6*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)
*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)+256*EllipticF((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^
(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2))*c^2*d^5*e*(-a*c)^(1/2)*(-(e*x+d)/(-c*d+(-a*c)^(1/2)
*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(
1/2)-72*EllipticE((-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2),(-(-c*d+(-a*c)^(1/2)*e)/(c*d+(-a*c)^(1/2)*e))^(1/2)
)*a^2*c*d^2*e^4*(-(e*x+d)/(-c*d+(-a*c)^(1/2)*e)*c)^(1/2)*((-c*x+(-a*c)^(1/2))/(c*d+(-a*c)^(1/2)*e)*e)^(1/2)*((
c*x+(-a*c)^(1/2))/(-c*d+(-a*c)^(1/2)*e)*e)^(1/2)-50*a^2*c*d*e^5*x)/(c*x^2+a)^(1/2)/(e*x+d)^(5/2)/e^7
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x^2+a)^(5/2)/(e*x+d)^(7/2),x, algorithm="maxima")
[Out]
integrate((c*x^2 + a)^(5/2)/(e*x + d)^(7/2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,x^2+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + c*x^2)^(5/2)/(d + e*x)^(7/2),x)
[Out]
int((a + c*x^2)^(5/2)/(d + e*x)^(7/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c*x**2+a)**(5/2)/(e*x+d)**(7/2),x)
[Out]
Integral((a + c*x**2)**(5/2)/(d + e*x)**(7/2), x)
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