3.2461 \(\int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx\)

Optimal. Leaf size=600 \[ -\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c} \]

[Out]

12/35*e*(-b*e+2*c*d)*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2+2/7*e*(e*x+d)^(5/2)*(c*x^2+b*x+a)^(1/2)/c+2/105*e*(
71*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*b*d))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3+8/105*(-b*e+2*c*d)*(11*c^2*d^
2+6*b^2*e^2-c*e*(13*a*e+11*b*d))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2)
,(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-
c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^4/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)
-2/105*(a*e^2-b*d*e+c*d^2)*(71*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+71*b*d))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/
2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)
/c^4/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]  time = 0.85, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {742, 832, 843, 718, 424, 419} \[ \frac {2 e \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right )}{105 c^3}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-c e (25 a e+71 b d)+24 b^2 e^2+71 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (13 a e+11 b d)+6 b^2 e^2+11 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {12 e (d+e x)^{3/2} \sqrt {a+b x+c x^2} (2 c d-b e)}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(105*c^3) + (12*e*
(2*c*d - b*e)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(35*c^2) + (2*e*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7
*c) + (8*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(11*c^2*d^2 + 6*b^2*e^2 - c*e*(11*b*d + 13*a*e))*Sqrt[d + e*x
]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*Sqrt[(c*(d + e*x))
/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a
*e^2)*(71*c^2*d^2 + 24*b^2*e^2 - c*e*(71*b*d + 25*a*e))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)
]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2
- 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(105*c^4*Sqrt[d + e*x]*Sqrt
[a + b*x + 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 718

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

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 832

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

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^{7/2}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {2 \int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (7 c d^2-e (b d+5 a e)\right )+3 e (2 c d-b e) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{7 c}\\ &=\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {4 \int \frac {\sqrt {d+e x} \left (\frac {1}{4} \left (35 c^2 d^3+6 b e^2 (b d+3 a e)-c d e (17 b d+61 a e)\right )+\frac {1}{4} e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) x\right )}{\sqrt {a+b x+c x^2}} \, dx}{35 c^2}\\ &=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \int \frac {\frac {1}{8} \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )+e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3}\\ &=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {\left (8 (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^3}+\frac {\left (8 \left (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac {1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^3 e}\\ &=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^4 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}+\frac {\left (16 \sqrt {2} \sqrt {b^2-4 a c} \left (-d e (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right )+\frac {1}{8} e \left (105 c^3 d^4-24 b^2 e^3 (b d+a e)-2 c^2 d^2 e (61 b d+127 a e)+c e^2 \left (89 b^2 d^2+150 a b d e+25 a^2 e^2\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^4 e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 e \left (71 c^2 d^2+24 b^2 e^2-c e (71 b d+25 a e)\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{105 c^3}+\frac {12 e (2 c d-b e) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}{35 c^2}+\frac {2 e (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c}+\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (11 c^2 d^2+6 b^2 e^2-c e (11 b d+13 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (71 c^2 d^2-71 b c d e+24 b^2 e^2-25 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^4 \sqrt {d+e x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 13.34, size = 5340, normalized size = 8.90 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt {e x + d}}{\sqrt {c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(e*x + d)/sqrt(c*x^2 + b*x + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x + a), x)

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maple [B]  time = 0.13, size = 6947, normalized size = 11.58 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {c x^{2} + b x + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(7/2)/sqrt(c*x^2 + b*x + a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^(1/2),x)

[Out]

int((d + e*x)^(7/2)/(a + b*x + c*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d + e x\right )^{\frac {7}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((d + e*x)**(7/2)/sqrt(a + b*x + c*x**2), x)

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