3.2544 \(\int \frac {d+e x}{(a+b x+c x^2)^{5/4}} \, dx\)

Optimal. Leaf size=490 \[ \frac {\sqrt {2} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac {2 \sqrt {2} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac {4 (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}-\frac {4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

[Out]

-4*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/4)+4*(-b*e+2*c*d)*(2*c*x+b)*(c*x^2+b*x+a)^(1/4)/(-
4*a*c+b^2)^(3/2)/c^(1/2)/(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))-2*(-b*e+2*c*d)*(cos(2*arctan(c^(
1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2
)/(-4*a*c+b^2)^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4))),1/2*2^(
1/2))*2^(1/2)*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))*((2*c*x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c*x
^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/c^(3/4)/(-4*a*c+b^2)^(1/4)/(2*c*x+b)+(-b*e+2*c*d)*(cos(2*arctan(c
^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1
/2)/(-4*a*c+b^2)^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*(c*x^2+b*x+a)^(1/4)*2^(1/2)/(-4*a*c+b^2)^(1/4))),1/2*2
^(1/2))*2^(1/2)*(1+2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))*((2*c*x+b)^2/(-4*a*c+b^2)/(1+2*c^(1/2)*(c
*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))^2)^(1/2)/c^(3/4)/(-4*a*c+b^2)^(1/4)/(2*c*x+b)

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Rubi [A]  time = 0.40, antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {638, 623, 305, 220, 1196} \[ \frac {\sqrt {2} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}-\frac {2 \sqrt {2} \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) (2 c d-b e) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac {4 (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e)}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )}-\frac {4 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(a + b*x + c*x^2)^(5/4),x]

[Out]

(-4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^(1/4)) + (4*(2*c*d - b*e)*(b + 2*c*x)*(a
+ b*x + c*x^2)^(1/4))/(Sqrt[c]*(b^2 - 4*a*c)^(3/2)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]))
- (2*Sqrt[2]*(2*c*d - b*e)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2
- 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a
+ b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(c^(3/4)*(b^2 - 4*a*c)^(1/4)*(b + 2*c*x)) + (Sqrt[2]*(2*c*d
- b*e)*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4
))/(b^2 - 4*a*c)^(1/4)], 1/2])/(c^(3/4)*(b^2 - 4*a*c)^(1/4)*(b + 2*c*x))

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin {align*} \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/4}} \, dx &=-\frac {4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac {(2 (2 c d-b e)) \int \frac {1}{\sqrt [4]{a+b x+c x^2}} \, dx}{b^2-4 a c}\\ &=-\frac {4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\left (8 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}\\ &=-\frac {4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\left (4 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\sqrt {c} \sqrt {b^2-4 a c} (b+2 c x)}-\frac {\left (4 (2 c d-b e) \sqrt {(b+2 c x)^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {2 \sqrt {c} x^2}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{\sqrt {c} \sqrt {b^2-4 a c} (b+2 c x)}\\ &=-\frac {4 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2}}+\frac {4 (2 c d-b e) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{\sqrt {c} \left (b^2-4 a c\right )^{3/2} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}-\frac {2 \sqrt {2} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}+\frac {\sqrt {2} (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )^2}} \left (1+\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{c^{3/4} \sqrt [4]{b^2-4 a c} (b+2 c x)}\\ \end {align*}

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Mathematica [C]  time = 0.22, size = 167, normalized size = 0.34 \[ -\frac {2 \left (2^{3/4} \left (-\sqrt {b^2-4 a c}+b+2 c x\right ) \sqrt [4]{\frac {\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}} (b e-2 c d) \, _2F_1\left (\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {-b-2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right )+6 c (-2 a e+b (d-e x)+2 c d x)\right )}{3 c \left (b^2-4 a c\right ) \sqrt [4]{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(a + b*x + c*x^2)^(5/4),x]

[Out]

(-2*(6*c*(-2*a*e + 2*c*d*x + b*(d - e*x)) + 2^(3/4)*(-2*c*d + b*e)*(b - Sqrt[b^2 - 4*a*c] + 2*c*x)*((b + Sqrt[
b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c])^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (-b + Sqrt[b^2 - 4*a*c] - 2*c*
x)/(2*Sqrt[b^2 - 4*a*c])]))/(3*c*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/4))

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fricas [F]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c x^{2} + b x + a\right )}^{\frac {3}{4}} {\left (e x + d\right )}}{c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")

[Out]

integral((c*x^2 + b*x + a)^(3/4)*(e*x + d)/(c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="giac")

[Out]

integrate((e*x + d)/(c*x^2 + b*x + a)^(5/4), x)

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maple [F]  time = 1.88, size = 0, normalized size = 0.00 \[ \int \frac {e x +d}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(c*x^2+b*x+a)^(5/4),x)

[Out]

int((e*x+d)/(c*x^2+b*x+a)^(5/4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e x + d}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")

[Out]

integrate((e*x + d)/(c*x^2 + b*x + a)^(5/4), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {d+e\,x}{{\left (c\,x^2+b\,x+a\right )}^{5/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(a + b*x + c*x^2)^(5/4),x)

[Out]

int((d + e*x)/(a + b*x + c*x^2)^(5/4), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d + e x}{\left (a + b x + c x^{2}\right )^{\frac {5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(c*x**2+b*x+a)**(5/4),x)

[Out]

Integral((d + e*x)/(a + b*x + c*x**2)**(5/4), x)

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