3.16.94 \(\int (d+e x)^3 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=328 \[ -\frac {7 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac {7 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d} \]

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Rubi [A]  time = 0.28, antiderivative size = 328, 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 = {670, 640, 612, 621, 206} \begin {gather*} \frac {7 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac {7 \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}}{48 c^3 d^3}+\frac {7 (d+e x) \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}}{40 c^2 d^2}-\frac {7 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac {(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(7*(c*d^2 - a*e^2)^3*(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])/(128*c^4*d^4*e)
+ (7*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(48*c^3*d^3) + (7*(c*d^2 - a*e^2)*(d + e
*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(40*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2)^(3/2))/(5*c*d) - (7*(c*d^2 - a*e^2)^5*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])])/(256*c^(9/2)*d^(9/2)*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)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {(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}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )\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}{10 d}\\ &=\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(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}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{16 d^2}\\ &=\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(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}}{5 c d}+\frac {\left (7 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{32 d^3}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(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}}{5 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^5\right ) \int \frac {1}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{256 c^4 d^4 e}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(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}}{5 c d}-\frac {\left (7 \left (c d^2-a e^2\right )^5\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 )}{128 c^4 d^4 e}\\ &=\frac {7 \left (c d^2-a e^2\right )^3 \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}}{128 c^4 d^4 e}+\frac {7 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac {7 \left (c d^2-a e^2\right ) (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac {(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}}{5 c d}-\frac {7 \left (c d^2-a e^2\right )^5 \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 )}{256 c^{9/2} d^{9/2} e^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 1.91, size = 291, normalized size = 0.89 \begin {gather*} \frac {(a e+c d x) \sqrt {(d+e x) (a e+c d x)} \left (-\frac {105 c^{5/2} d^{5/2} \sqrt {c d} \left (c d^2-a e^2\right )^{9/2} \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}}}+336 c^5 d^5 (d+e x)^2 \left (c d^2-a e^2\right )+280 c^4 d^4 (d+e x) \left (c d^2-a e^2\right )^2+\frac {105 c^3 d^3 \left (c d^2-a e^2\right )^4}{e (a e+c d x)}+210 \left (c^2 d^3-a c d e^2\right )^3+384 c^6 d^6 (d+e x)^3\right )}{1920 c^7 d^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*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)]*(210*(c^2*d^3 - a*c*d*e^2)^3 + (105*c^3*d^3*(c*d^2 - a*e^2)^4)/(e
*(a*e + c*d*x)) + 280*c^4*d^4*(c*d^2 - a*e^2)^2*(d + e*x) + 336*c^5*d^5*(c*d^2 - a*e^2)*(d + e*x)^2 + 384*c^6*
d^6*(d + e*x)^3 - (105*c^(5/2)*d^(5/2)*Sqrt[c*d]*(c*d^2 - a*e^2)^(9/2)*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)])))/(1920*c^7*d^7)

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IntegrateAlgebraic [F]  time = 181.11, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

$Aborted

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fricas [A]  time = 0.47, size = 844, normalized size = 2.57 \begin {gather*} \left [\frac {105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} + 105 \, c^{5} d^{9} e + 790 \, a c^{4} d^{7} e^{3} - 896 \, a^{2} c^{3} d^{5} e^{5} + 490 \, a^{3} c^{2} d^{3} e^{7} - 105 \, a^{4} c d e^{9} + 48 \, {\left (31 \, c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (263 \, c^{5} d^{7} e^{3} + 32 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (605 \, c^{5} d^{8} e^{2} + 289 \, a c^{4} d^{6} e^{4} - 161 \, a^{2} c^{3} d^{4} e^{6} + 35 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{7680 \, c^{5} d^{5} e^{2}}, \frac {105 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 (384 \, c^{5} d^{5} e^{5} x^{4} + 105 \, c^{5} d^{9} e + 790 \, a c^{4} d^{7} e^{3} - 896 \, a^{2} c^{3} d^{5} e^{5} + 490 \, a^{3} c^{2} d^{3} e^{7} - 105 \, a^{4} c d e^{9} + 48 \, {\left (31 \, c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{3} + 8 \, {\left (263 \, c^{5} d^{7} e^{3} + 32 \, a c^{4} d^{5} e^{5} - 7 \, a^{2} c^{3} d^{3} e^{7}\right )} x^{2} + 2 \, {\left (605 \, c^{5} d^{8} e^{2} + 289 \, a c^{4} d^{6} e^{4} - 161 \, a^{2} c^{3} d^{4} e^{6} + 35 \, a^{3} c^{2} d^{2} e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{3840 \, c^{5} d^{5} e^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="fricas")

[Out]

[1/7680*(105*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^1
0)*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*(384*c^5*d^5*e^5*x^4 + 1
05*c^5*d^9*e + 790*a*c^4*d^7*e^3 - 896*a^2*c^3*d^5*e^5 + 490*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9 + 48*(31*c^5*d^
6*e^4 + a*c^4*d^4*e^6)*x^3 + 8*(263*c^5*d^7*e^3 + 32*a*c^4*d^5*e^5 - 7*a^2*c^3*d^3*e^7)*x^2 + 2*(605*c^5*d^8*e
^2 + 289*a*c^4*d^6*e^4 - 161*a^2*c^3*d^4*e^6 + 35*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x))/(c^5*d^5*e^2), 1/3840*(105*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*
c*d^2*e^8 - a^5*e^10)*sqrt(-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*(384*c^5*d^5*e^5*x^4 + 1
05*c^5*d^9*e + 790*a*c^4*d^7*e^3 - 896*a^2*c^3*d^5*e^5 + 490*a^3*c^2*d^3*e^7 - 105*a^4*c*d*e^9 + 48*(31*c^5*d^
6*e^4 + a*c^4*d^4*e^6)*x^3 + 8*(263*c^5*d^7*e^3 + 32*a*c^4*d^5*e^5 - 7*a^2*c^3*d^3*e^7)*x^2 + 2*(605*c^5*d^8*e
^2 + 289*a*c^4*d^6*e^4 - 161*a^2*c^3*d^4*e^6 + 35*a^3*c^2*d^2*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)
*x))/(c^5*d^5*e^2)]

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giac [A]  time = 0.42, size = 377, normalized size = 1.15 \begin {gather*} \frac {1}{1920} \, \sqrt {c d x^{2} e + c d^{2} x + a x e^{2} + a d e} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x e^{3} + \frac {{\left (31 \, c^{4} d^{5} e^{6} + a c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (263 \, c^{4} d^{6} e^{5} + 32 \, a c^{3} d^{4} e^{7} - 7 \, a^{2} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (605 \, c^{4} d^{7} e^{4} + 289 \, a c^{3} d^{5} e^{6} - 161 \, a^{2} c^{2} d^{3} e^{8} + 35 \, a^{3} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac {{\left (105 \, c^{4} d^{8} e^{3} + 790 \, a c^{3} d^{6} e^{5} - 896 \, a^{2} c^{2} d^{4} e^{7} + 490 \, a^{3} c d^{2} e^{9} - 105 \, a^{4} e^{11}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} + \frac {7 \, {\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\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 )}{256 \, \sqrt {c d} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x, algorithm="giac")

[Out]

1/1920*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(6*(8*x*e^3 + (31*c^4*d^5*e^6 + a*c^3*d^3*e^8)*e^(-4)
/(c^4*d^4))*x + (263*c^4*d^6*e^5 + 32*a*c^3*d^4*e^7 - 7*a^2*c^2*d^2*e^9)*e^(-4)/(c^4*d^4))*x + (605*c^4*d^7*e^
4 + 289*a*c^3*d^5*e^6 - 161*a^2*c^2*d^3*e^8 + 35*a^3*c*d*e^10)*e^(-4)/(c^4*d^4))*x + (105*c^4*d^8*e^3 + 790*a*
c^3*d^6*e^5 - 896*a^2*c^2*d^4*e^7 + 490*a^3*c*d^2*e^9 - 105*a^4*e^11)*e^(-4)/(c^4*d^4)) + 7/256*(c^5*d^10 - 5*
a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*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^4*d^4)

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maple [B]  time = 0.09, size = 968, normalized size = 2.95 \begin {gather*} \frac {7 a^{5} e^{9} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{256 \sqrt {c d e}\, c^{4} d^{4}}-\frac {35 a^{4} e^{7} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{256 \sqrt {c d e}\, c^{3} d^{2}}+\frac {35 a^{3} e^{5} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{128 \sqrt {c d e}\, c^{2}}-\frac {35 a^{2} d^{2} e^{3} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{128 \sqrt {c d e}\, c}+\frac {35 a \,d^{4} e \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{256 \sqrt {c d e}}-\frac {7 c \,d^{6} \ln \left (\frac {c d e x +\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}}{\sqrt {c d e}}+\sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\right )}{256 \sqrt {c d e}\, e}-\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{3} e^{6} x}{64 c^{3} d^{3}}+\frac {21 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{2} e^{4} x}{64 c^{2} d}-\frac {21 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a d \,e^{2} x}{64 c}+\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{3} x}{64}-\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{4} e^{7}}{128 c^{4} d^{4}}+\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a^{3} e^{5}}{64 c^{3} d^{2}}-\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, a \,d^{2} e}{64 c}+\frac {7 \sqrt {c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x}\, d^{4}}{128 e}+\frac {\left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} e^{2} x^{2}}{5 c d}-\frac {7 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} a \,e^{3} x}{40 c^{2} d^{2}}+\frac {23 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} e x}{40 c}+\frac {7 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} a^{2} e^{4}}{48 c^{3} d^{3}}-\frac {7 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} a \,e^{2}}{15 c^{2} d}+\frac {25 \left (c d e \,x^{2}+a d e +\left (a \,e^{2}+c \,d^{2}\right ) x \right )^{\frac {3}{2}} d}{48 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2),x)

[Out]

7/128/e*d^4*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+25/48/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+7/64*d^3
*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+23/40*e/c*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+35/128*e^5/c^2*
ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^3+1/5*
e^2*x^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)/d/c+21/64*e^4/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*
a^2-7/64*e^6/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-21/64*e^2/c*d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(1/2)*x*a+7/256*e^9/c^4/d^4*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)
*x)^(1/2))/(c*d*e)^(1/2)*a^5-7/15*e^2/c^2/d*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-7/128*e^7/c^4/d^4*(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-35/256*e^7/c^3/d^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)*a^4-7/40*e^3/c^2/d^2*x*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2
)*a-35/128*e^3/c*d^2*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2))/(
c*d*e)^(1/2)*a^2+35/256*e*d^4*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)
^(1/2))/(c*d*e)^(1/2)*a+7/64*e^5/c^3/d^2*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3-7/64*e/c*d^2*(c*d*e*x^2+a
*d*e+(a*e^2+c*d^2)*x)^(1/2)*a-7/256/e*c*d^6*ln((c*d*e*x+1/2*a*e^2+1/2*c*d^2)/(c*d*e)^(1/2)+(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(1/2))/(c*d*e)^(1/2)+7/48*e^4/c^3/d^3*(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2

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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)^3*(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 \begin {gather*} \int {\left (d+e\,x\right )}^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^3*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2),x)

[Out]

int((d + e*x)^3*(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 \begin {gather*} \int \sqrt {\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(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)**3, x)

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