3.20.64 \(\int \frac {(d+e x)^6}{(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}} \, dx\) [1964]

Optimal. Leaf size=294 \[ -\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}} \]

[Out]

-2/3*(e*x+d)^5/c/d/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+35/8*e^(3/2)*(-a*e^2+c*d^2)^2*arctanh(1/2*(2*c*d*e*
x+a*e^2+c*d^2)/c^(1/2)/d^(1/2)/e^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(9/2)/d^(9/2)-14/3*e*(e*x+d)
^3/c^2/d^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/4*e^2*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
1/2)/c^4/d^4+35/6*e^2*(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^3/d^3

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Rubi [A]
time = 0.15, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {682, 684, 654, 635, 212} \begin {gather*} \frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{6 c^3 d^3}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac {2 (d+e x)^5}{3 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(-2*(d + e*x)^5)/(3*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (14*e*(d + e*x)^3)/(3*c^2*d^2*Sqrt[a*
d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (35*e^2*(c*d^2 - a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(
4*c^4*d^4) + (35*e^2*(d + e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(6*c^3*d^3) + (35*e^(3/2)*(c*d^2 -
a*e^2)^2*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2])])/(8*c^(9/2)*d^(9/2))

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 635

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

Rule 654

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

Rule 682

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*(p + 1))), x] - Dist[e^2*((m + p)/(c*(p + 1))), Int[(d + e*x)^(m - 2)*(a + b*x +
c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]

Rule 684

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[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1))), Int[(d + e
*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {(d+e x)^6}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}} \, dx &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}+\frac {(7 e) \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{3 c d}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^2}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 c^2 d^2}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )\right ) \int \frac {d+e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 c^3 d^3}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c^4 d^4}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {\left (35 e^2 \left (c d^2-a e^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{4 c d e-x^2} \, dx,x,\frac {c d^2+a e^2+2 c d e x}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^4 d^4}\\ &=-\frac {2 (d+e x)^5}{3 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}-\frac {14 e (d+e x)^3}{3 c^2 d^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {35 e^2 \left (c d^2-a e^2\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^4 d^4}+\frac {35 e^2 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{6 c^3 d^3}+\frac {35 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {c d^2+a e^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{9/2} d^{9/2}}\\ \end {align*}

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Mathematica [A]
time = 0.42, size = 212, normalized size = 0.72 \begin {gather*} \frac {\sqrt {(a e+c d x) (d+e x)} \left (-\sqrt {c} \sqrt {d} (a e+c d x) \left (105 e^3-\frac {175 c d e^2 (d+e x)}{a e+c d x}+\frac {56 c^2 d^2 e (d+e x)^2}{(a e+c d x)^2}+\frac {8 c^3 d^3 (d+e x)^3}{(a e+c d x)^3}\right )+\frac {105 e^{3/2} \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {e} \sqrt {a e+c d x}}\right )}{\sqrt {a e+c d x} \sqrt {d+e x}}\right )}{12 c^{9/2} d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*(-(Sqrt[c]*Sqrt[d]*(a*e + c*d*x)*(105*e^3 - (175*c*d*e^2*(d + e*x))/(a*e + c*d*
x) + (56*c^2*d^2*e*(d + e*x)^2)/(a*e + c*d*x)^2 + (8*c^3*d^3*(d + e*x)^3)/(a*e + c*d*x)^3)) + (105*e^(3/2)*(c*
d^2 - a*e^2)^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[a*e + c*d*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d
+ e*x])))/(12*c^(9/2)*d^(9/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(7419\) vs. \(2(260)=520\).
time = 0.72, size = 7420, normalized size = 25.24

method	result	size
default	\(\text {Expression too large to display}\)	\(7420\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d^2-%e^2*a>0)', see `assume?
` for more d

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Fricas [A]
time = 10.82, size = 822, normalized size = 2.80 \begin {gather*} \left [\frac {105 \, {\left (c^{4} d^{6} x^{2} e + 2 \, a c^{3} d^{5} x e^{2} - 4 \, a^{2} c^{2} d^{3} x e^{4} + 2 \, a^{3} c d x e^{6} + a^{4} e^{7} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{5} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{3}\right )} \sqrt {\frac {1}{c d}} e^{\frac {1}{2}} \log \left (8 \, c^{2} d^{3} x e + c^{2} d^{4} + 8 \, a c d x e^{3} + a^{2} e^{4} + 4 \, {\left (2 \, c^{2} d^{2} x e + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} \sqrt {\frac {1}{c d}} e^{\frac {1}{2}} + 2 \, {\left (4 \, c^{2} d^{2} x^{2} + 3 \, a c d^{2}\right )} e^{2}\right ) - 4 \, {\left (80 \, c^{3} d^{5} x e + 8 \, c^{3} d^{6} + 140 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} + 7 \, {\left (3 \, a c^{2} d^{2} x^{2} - 25 \, a^{2} c d^{2}\right )} e^{4} - 2 \, {\left (3 \, c^{3} d^{3} x^{3} + 119 \, a c^{2} d^{3} x\right )} e^{3} - {\left (39 \, c^{3} d^{4} x^{2} - 56 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{48 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} x e + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (c^{4} d^{6} x^{2} e + 2 \, a c^{3} d^{5} x e^{2} - 4 \, a^{2} c^{2} d^{3} x e^{4} + 2 \, a^{3} c d x e^{6} + a^{4} e^{7} + {\left (a^{2} c^{2} d^{2} x^{2} - 2 \, a^{3} c d^{2}\right )} e^{5} - {\left (2 \, a c^{3} d^{4} x^{2} - a^{2} c^{2} d^{4}\right )} e^{3}\right )} \sqrt {-\frac {e}{c d}} \arctan \left (\frac {\sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (2 \, c d x e + c d^{2} + a e^{2}\right )} \sqrt {-\frac {e}{c d}}}{2 \, {\left (c d^{2} x e + a x e^{3} + {\left (c d x^{2} + a d\right )} e^{2}\right )}}\right ) + 2 \, {\left (80 \, c^{3} d^{5} x e + 8 \, c^{3} d^{6} + 140 \, a^{2} c d x e^{5} + 105 \, a^{3} e^{6} + 7 \, {\left (3 \, a c^{2} d^{2} x^{2} - 25 \, a^{2} c d^{2}\right )} e^{4} - 2 \, {\left (3 \, c^{3} d^{3} x^{3} + 119 \, a c^{2} d^{3} x\right )} e^{3} - {\left (39 \, c^{3} d^{4} x^{2} - 56 \, a c^{2} d^{4}\right )} e^{2}\right )} \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} x e + a^{2} c^{4} d^{4} e^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="fricas")

[Out]

[1/48*(105*(c^4*d^6*x^2*e + 2*a*c^3*d^5*x*e^2 - 4*a^2*c^2*d^3*x*e^4 + 2*a^3*c*d*x*e^6 + a^4*e^7 + (a^2*c^2*d^2
*x^2 - 2*a^3*c*d^2)*e^5 - (2*a*c^3*d^4*x^2 - a^2*c^2*d^4)*e^3)*sqrt(1/(c*d))*e^(1/2)*log(8*c^2*d^3*x*e + c^2*d
^4 + 8*a*c*d*x*e^3 + a^2*e^4 + 4*(2*c^2*d^2*x*e + c^2*d^3 + a*c*d*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d
)*e)*sqrt(1/(c*d))*e^(1/2) + 2*(4*c^2*d^2*x^2 + 3*a*c*d^2)*e^2) - 4*(80*c^3*d^5*x*e + 8*c^3*d^6 + 140*a^2*c*d*
x*e^5 + 105*a^3*e^6 + 7*(3*a*c^2*d^2*x^2 - 25*a^2*c*d^2)*e^4 - 2*(3*c^3*d^3*x^3 + 119*a*c^2*d^3*x)*e^3 - (39*c
^3*d^4*x^2 - 56*a*c^2*d^4)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^6*d^6*x^2 + 2*a*c^5*d^5*x*e +
a^2*c^4*d^4*e^2), -1/24*(105*(c^4*d^6*x^2*e + 2*a*c^3*d^5*x*e^2 - 4*a^2*c^2*d^3*x*e^4 + 2*a^3*c*d*x*e^6 + a^4*
e^7 + (a^2*c^2*d^2*x^2 - 2*a^3*c*d^2)*e^5 - (2*a*c^3*d^4*x^2 - a^2*c^2*d^4)*e^3)*sqrt(-e/(c*d))*arctan(1/2*sqr
t(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(2*c*d*x*e + c*d^2 + a*e^2)*sqrt(-e/(c*d))/(c*d^2*x*e + a*x*e^3 + (c*
d*x^2 + a*d)*e^2)) + 2*(80*c^3*d^5*x*e + 8*c^3*d^6 + 140*a^2*c*d*x*e^5 + 105*a^3*e^6 + 7*(3*a*c^2*d^2*x^2 - 25
*a^2*c*d^2)*e^4 - 2*(3*c^3*d^3*x^3 + 119*a*c^2*d^3*x)*e^3 - (39*c^3*d^4*x^2 - 56*a*c^2*d^4)*e^2)*sqrt(c*d^2*x
+ a*x*e^2 + (c*d*x^2 + a*d)*e))/(c^6*d^6*x^2 + 2*a*c^5*d^5*x*e + a^2*c^4*d^4*e^2)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d + e x\right )^{6}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2),x)

[Out]

Integral((d + e*x)**6/((d + e*x)*(a*e + c*d*x))**(5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (263) = 526\).
time = 1.51, size = 798, normalized size = 2.71 \begin {gather*} \frac {1}{4} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (\frac {2 \, x e^{3}}{c^{3} d^{3}} + \frac {{\left (13 \, c^{8} d^{9} e^{3} - 11 \, a c^{7} d^{7} e^{5}\right )} e^{\left (-1\right )}}{c^{11} d^{11}}\right )} - \frac {35 \, {\left (\sqrt {c d} c^{2} d^{4} e^{\frac {5}{2}} - 2 \, \sqrt {c d} a c d^{2} e^{\frac {9}{2}} + \sqrt {c d} a^{2} e^{\frac {13}{2}}\right )} e^{\left (-1\right )} \log \left ({\left | -\sqrt {c d} c d^{2} e^{\frac {1}{2}} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{8 \, c^{5} d^{5}} + \frac {2 \, {\left (c^{5} d^{10} + 3 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} c^{4} d^{8} e^{\frac {1}{2}} - 2 \, a c^{4} d^{8} e^{2} + 12 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} c^{4} d^{7} e + 12 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} a c^{3} d^{6} e^{\frac {5}{2}} + 10 \, a^{2} c^{3} d^{6} e^{4} - 36 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} a c^{3} d^{5} e^{3} - 54 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} a^{2} c^{2} d^{4} e^{\frac {9}{2}} - 28 \, a^{3} c^{2} d^{4} e^{6} + 36 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} a^{2} c^{2} d^{3} e^{5} + 60 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} a^{3} c d^{2} e^{\frac {13}{2}} + 29 \, a^{4} c d^{2} e^{8} - 12 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )}^{2} a^{3} c d e^{7} - 21 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} a^{4} e^{\frac {17}{2}} - 10 \, a^{5} e^{10}\right )}}{3 \, {\left ({\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d + \sqrt {c d} a e^{\frac {3}{2}}\right )}^{3} c^{3} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^6/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2),x, algorithm="giac")

[Out]

1/4*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*x*e^3/(c^3*d^3) + (13*c^8*d^9*e^3 - 11*a*c^7*d^7*e^5)*e^(-1
)/(c^11*d^11)) - 35/8*(sqrt(c*d)*c^2*d^4*e^(5/2) - 2*sqrt(c*d)*a*c*d^2*e^(9/2) + sqrt(c*d)*a^2*e^(13/2))*e^(-1
)*log(abs(-sqrt(c*d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d
*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5) + 2/3*(c^5*d^10 + 3*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x
*e^2 + a*d*e))*sqrt(c*d)*c^4*d^8*e^(1/2) - 2*a*c^4*d^8*e^2 + 12*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*
x + a*x*e^2 + a*d*e))^2*c^4*d^7*e + 12*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqr
t(c*d)*a*c^3*d^6*e^(5/2) + 10*a^2*c^3*d^6*e^4 - 36*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))^2*a*c^3*d^5*e^3 - 54*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c*d)*a^
2*c^2*d^4*e^(9/2) - 28*a^3*c^2*d^4*e^6 + 36*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)
)^2*a^2*c^2*d^3*e^5 + 60*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c*d)*a^3*c*d
^2*e^(13/2) + 29*a^4*c*d^2*e^8 - 12*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))^2*a^3*
c*d*e^7 - 21*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*sqrt(c*d)*a^4*e^(17/2) - 10*a
^5*e^10)/(((sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e))*c*d + sqrt(c*d)*a*e^(3/2))^3*c^
3*d^3)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^6}{{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2),x)

[Out]

int((d + e*x)^6/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(5/2), x)

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