3.1907 \(\int (d+e x)^4 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=388 \[ -\frac {21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac {21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]

[Out]

7/64*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4+21/160*(-a*e^2+c*d^2)^2*(e*x+d)*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3+3/20*(-a*e^2+c*d^2)*(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/
c^2/d^2+1/6*(e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d-21/1024*(-a*e^2+c*d^2)^6*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^(11/2)/d^(11/2)/e^(3/2)+21
/512*(-a*e^2+c*d^2)^4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^5/d^5/e

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Rubi [A]  time = 0.44, antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {670, 640, 612, 621, 206} \[ \frac {21 \left (c d^2-a e^2\right )^4 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 (d+e x) \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 (d+e x)^2 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{20 c^2 d^2}-\frac {21 \left (c d^2-a e^2\right )^6 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{11/2} d^{11/2} e^{3/2}}+\frac {(d+e x)^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(c*d^2 - a*e^2)^4*(c*d^2 + a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^5*d^5*e)
+ (7*(c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(64*c^4*d^4) + (21*(c*d^2 - a*e^2)^2*(d
+ e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(160*c^3*d^3) + (3*(c*d^2 - a*e^2)*(d + e*x)^2*(a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(20*c^2*d^2) + ((d + e*x)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/
2))/(6*c*d) - (21*(c*d^2 - a*e^2)^6*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])])/(1024*c^(11/2)*d^(11/2)*e^(3/2))

Rule 206

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

Rule 612

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

Rule 621

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 640

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 670

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 (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (3 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{4 d}\\ &=\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{40 d^2}\\ &=\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{64 d^3}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}+\frac {\left (21 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{128 d^4}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {\left (21 \left (c d^2-a e^2\right )^6\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{1024 c^5 d^5 e}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {\left (21 \left (c d^2-a e^2\right )^6\right ) \operatorname {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 )}{512 c^5 d^5 e}\\ &=\frac {21 \left (c d^2-a e^2\right )^4 \left (c d^2+a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{512 c^5 d^5 e}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{64 c^4 d^4}+\frac {21 \left (c d^2-a e^2\right )^2 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{160 c^3 d^3}+\frac {3 \left (c d^2-a e^2\right ) (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{20 c^2 d^2}+\frac {(d+e x)^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{6 c d}-\frac {21 \left (c d^2-a e^2\right )^6 \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 )}{1024 c^{11/2} d^{11/2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 2.43, size = 320, normalized size = 0.82 \[ \frac {(a e+c d x) \sqrt {(d+e x) (a e+c d x)} \left (-\frac {315 c^{7/2} d^{7/2} \sqrt {c d} \left (c d^2-a e^2\right )^{11/2} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d} \sqrt {c d^2-a e^2}}\right )}{e^{3/2} (a e+c d x)^{3/2} \sqrt {\frac {c d (d+e x)}{c d^2-a e^2}}}+1152 c^7 d^7 (d+e x)^3 \left (c d^2-a e^2\right )+1008 c^6 d^6 (d+e x)^2 \left (c d^2-a e^2\right )^2+840 c^5 d^5 (d+e x) \left (c d^2-a e^2\right )^3+\frac {315 c^4 d^4 \left (c d^2-a e^2\right )^5}{e (a e+c d x)}+630 \left (c^2 d^3-a c d e^2\right )^4+1280 c^8 d^8 (d+e x)^4\right )}{7680 c^9 d^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*e + c*d*x)*Sqrt[(a*e + c*d*x)*(d + e*x)]*(630*(c^2*d^3 - a*c*d*e^2)^4 + (315*c^4*d^4*(c*d^2 - a*e^2)^5)/(e
*(a*e + c*d*x)) + 840*c^5*d^5*(c*d^2 - a*e^2)^3*(d + e*x) + 1008*c^6*d^6*(c*d^2 - a*e^2)^2*(d + e*x)^2 + 1152*
c^7*d^7*(c*d^2 - a*e^2)*(d + e*x)^3 + 1280*c^8*d^8*(d + e*x)^4 - (315*c^(7/2)*d^(7/2)*Sqrt[c*d]*(c*d^2 - a*e^2
)^(11/2)*ArcSinh[(Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x])/(Sqrt[c*d]*Sqrt[c*d^2 - a*e^2])])/(e^(3/2)*(a*e +
c*d*x)^(3/2)*Sqrt[(c*d*(d + e*x))/(c*d^2 - a*e^2)])))/(7680*c^9*d^9)

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fricas [A]  time = 1.25, size = 1040, normalized size = 2.68 \[ \left [\frac {315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e + 3335 \, a c^{5} d^{9} e^{3} - 5058 \, a^{2} c^{4} d^{7} e^{5} + 4158 \, a^{3} c^{3} d^{5} e^{7} - 1785 \, a^{4} c^{2} d^{3} e^{9} + 315 \, a^{5} c d e^{11} + 128 \, {\left (49 \, c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (759 \, c^{6} d^{8} e^{4} + 50 \, a c^{5} d^{6} e^{6} - 9 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (1429 \, c^{6} d^{9} e^{3} + 267 \, a c^{5} d^{7} e^{5} - 117 \, a^{2} c^{4} d^{5} e^{7} + 21 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (2455 \, c^{6} d^{10} e^{2} + 1612 \, a c^{5} d^{8} e^{4} - 1350 \, a^{2} c^{4} d^{6} e^{6} + 588 \, a^{3} c^{3} d^{4} e^{8} - 105 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{30720 \, c^{6} d^{6} e^{2}}, \frac {315 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (1280 \, c^{6} d^{6} e^{6} x^{5} + 315 \, c^{6} d^{11} e + 3335 \, a c^{5} d^{9} e^{3} - 5058 \, a^{2} c^{4} d^{7} e^{5} + 4158 \, a^{3} c^{3} d^{5} e^{7} - 1785 \, a^{4} c^{2} d^{3} e^{9} + 315 \, a^{5} c d e^{11} + 128 \, {\left (49 \, c^{6} d^{7} e^{5} + a c^{5} d^{5} e^{7}\right )} x^{4} + 16 \, {\left (759 \, c^{6} d^{8} e^{4} + 50 \, a c^{5} d^{6} e^{6} - 9 \, a^{2} c^{4} d^{4} e^{8}\right )} x^{3} + 8 \, {\left (1429 \, c^{6} d^{9} e^{3} + 267 \, a c^{5} d^{7} e^{5} - 117 \, a^{2} c^{4} d^{5} e^{7} + 21 \, a^{3} c^{3} d^{3} e^{9}\right )} x^{2} + 2 \, {\left (2455 \, c^{6} d^{10} e^{2} + 1612 \, a c^{5} d^{8} e^{4} - 1350 \, a^{2} c^{4} d^{6} e^{6} + 588 \, a^{3} c^{3} d^{4} e^{8} - 105 \, a^{4} c^{2} d^{2} e^{10}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{15360 \, c^{6} d^{6} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/30720*(315*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*
a^5*c*d^2*e^10 + a^6*e^12)*sqrt(c*d*e)*log(8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 4*
(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e + 3335*a*c^5*d^9*e^3 - 5058*a^2*c^4*d^7*e^5 + 4158*a^3*c^3*d^5*e^7 - 17
85*a^4*c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(49*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(759*c^6*d^8*e^4 + 50*a*
c^5*d^6*e^6 - 9*a^2*c^4*d^4*e^8)*x^3 + 8*(1429*c^6*d^9*e^3 + 267*a*c^5*d^7*e^5 - 117*a^2*c^4*d^5*e^7 + 21*a^3*
c^3*d^3*e^9)*x^2 + 2*(2455*c^6*d^10*e^2 + 1612*a*c^5*d^8*e^4 - 1350*a^2*c^4*d^6*e^6 + 588*a^3*c^3*d^4*e^8 - 10
5*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*e^2), 1/15360*(315*(c^6*d^12 - 6*
a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*s
qrt(-c*d*e)*arctan(1/2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(c
^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)) + 2*(1280*c^6*d^6*e^6*x^5 + 315*c^6*d^11*e + 3335*a
*c^5*d^9*e^3 - 5058*a^2*c^4*d^7*e^5 + 4158*a^3*c^3*d^5*e^7 - 1785*a^4*c^2*d^3*e^9 + 315*a^5*c*d*e^11 + 128*(49
*c^6*d^7*e^5 + a*c^5*d^5*e^7)*x^4 + 16*(759*c^6*d^8*e^4 + 50*a*c^5*d^6*e^6 - 9*a^2*c^4*d^4*e^8)*x^3 + 8*(1429*
c^6*d^9*e^3 + 267*a*c^5*d^7*e^5 - 117*a^2*c^4*d^5*e^7 + 21*a^3*c^3*d^3*e^9)*x^2 + 2*(2455*c^6*d^10*e^2 + 1612*
a*c^5*d^8*e^4 - 1350*a^2*c^4*d^6*e^6 + 588*a^3*c^3*d^4*e^8 - 105*a^4*c^2*d^2*e^10)*x)*sqrt(c*d*e*x^2 + a*d*e +
(c*d^2 + a*e^2)*x))/(c^6*d^6*e^2)]

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giac [A]  time = 0.49, size = 476, normalized size = 1.23 \[ \frac {1}{7680} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, x e^{4} + \frac {{\left (49 \, c^{5} d^{6} e^{8} + a c^{4} d^{4} e^{10}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (759 \, c^{5} d^{7} e^{7} + 50 \, a c^{4} d^{5} e^{9} - 9 \, a^{2} c^{3} d^{3} e^{11}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (1429 \, c^{5} d^{8} e^{6} + 267 \, a c^{4} d^{6} e^{8} - 117 \, a^{2} c^{3} d^{4} e^{10} + 21 \, a^{3} c^{2} d^{2} e^{12}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (2455 \, c^{5} d^{9} e^{5} + 1612 \, a c^{4} d^{7} e^{7} - 1350 \, a^{2} c^{3} d^{5} e^{9} + 588 \, a^{3} c^{2} d^{3} e^{11} - 105 \, a^{4} c d e^{13}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} x + \frac {{\left (315 \, c^{5} d^{10} e^{4} + 3335 \, a c^{4} d^{8} e^{6} - 5058 \, a^{2} c^{3} d^{6} e^{8} + 4158 \, a^{3} c^{2} d^{4} e^{10} - 1785 \, a^{4} c d^{2} e^{12} + 315 \, a^{5} e^{14}\right )} e^{\left (-5\right )}}{c^{5} d^{5}}\right )} + \frac {21 \, {\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -c d^{2} - 2 \, {\left (\sqrt {c d} x e^{\frac {1}{2}} - \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} \sqrt {c d} e^{\frac {1}{2}} - a e^{2} \right |}\right )}{1024 \, \sqrt {c d} c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

1/7680*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(2*(8*(10*x*e^4 + (49*c^5*d^6*e^8 + a*c^4*d^4*e^10)*e
^(-5)/(c^5*d^5))*x + (759*c^5*d^7*e^7 + 50*a*c^4*d^5*e^9 - 9*a^2*c^3*d^3*e^11)*e^(-5)/(c^5*d^5))*x + (1429*c^5
*d^8*e^6 + 267*a*c^4*d^6*e^8 - 117*a^2*c^3*d^4*e^10 + 21*a^3*c^2*d^2*e^12)*e^(-5)/(c^5*d^5))*x + (2455*c^5*d^9
*e^5 + 1612*a*c^4*d^7*e^7 - 1350*a^2*c^3*d^5*e^9 + 588*a^3*c^2*d^3*e^11 - 105*a^4*c*d*e^13)*e^(-5)/(c^5*d^5))*
x + (315*c^5*d^10*e^4 + 3335*a*c^4*d^8*e^6 - 5058*a^2*c^3*d^6*e^8 + 4158*a^3*c^2*d^4*e^10 - 1785*a^4*c*d^2*e^1
2 + 315*a^5*e^14)*e^(-5)/(c^5*d^5)) + 21/1024*(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d
^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*e^(-3/2)*log(abs(-c*d^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))*sqrt(c*d)*e^(1/2) - a*e^2))/(sqrt(c*d)*c^5*d^5)

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maple [B]  time = 0.12, size = 1327, normalized size = 3.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

107/192*d^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+21/256*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+21/
512/e*d^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+21/512*e^9/d^5/c^5*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
^5-63/512*e*d^3/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+21/256*e^5/d/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2)*a^3+21/256*e^3*d/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2+63/512*e*d^5*ln((1/2*a*e^2+1/2*c*d^2+c
*d*e*x)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a+63/128*e^4/c^2*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-63/512*e^7/d^3/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+149/160*e*d/c*x*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-237/320*e^2/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+13/20*e^2/c*x
^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+1/6*e^3*x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/c+21/160*e^5/
d^3/c^3*x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+147/320*e^4/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3
/2)*a^2-7/64*e^6/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-21/1024/e*d^7*c*ln((1/2*a*e^2+1/2*c*d^2+c
*d*e*x)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)-21/64*e^2*d^2/c*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*x*a-3/20*e^4/d^2/c^2*x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+105/256*e^5*d/c^2*ln
((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a^3-21/64*
e^6/d^2/c^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-21/1024*e^11/d^5/c^5*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x
)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a^6+63/512*e^9/d^3/c^4*ln((1/2*a*e^2+1/
2*c*d^2+c*d*e*x)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a^5-315/1024*e^7/d/c^3*l
n((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a^4+21/25
6*e^8/d^4/c^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4-315/1024*e^3*d^3/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x
)/(d*c*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*c*e)^(1/2)*a^2-9/16*e^3/d/c^2*x*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(3/2)*a

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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(a*e^2-c*d^2>0)', see `assume?`
for more details)Is a*e^2-c*d^2 zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)

[Out]

int((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**4, x)

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