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

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

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Rubi [A]  time = 0.21, 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 = {668, 670, 640, 621, 206} \begin {gather*} -\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 {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 {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^{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 {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 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 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 668

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 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 \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 ) \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 )}{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 [C]  time = 0.06, size = 112, normalized size = 0.38 \begin {gather*} -\frac {2 \left (c d^2-a e^2\right )^3 \sqrt {(d+e x) (a e+c d x)} \, _2F_1\left (-\frac {7}{2},-\frac {3}{2};-\frac {1}{2};\frac {e (a e+c d x)}{a e^2-c d^2}\right )}{3 c^4 d^4 (a e+c d x)^2 \sqrt {\frac {c d (d+e x)}{c d^2-a e^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]

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

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IntegrateAlgebraic [F]  time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 2.78, size = 833, normalized size = 2.83 \begin {gather*} \left [\frac {105 \, {\left (a^{2} c^{2} d^{4} e^{3} - 2 \, a^{3} c d^{2} e^{5} + a^{4} e^{7} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{2} - 2 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} x\right )} \sqrt {\frac {e}{c d}} \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} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \, {\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {\frac {e}{c d}}\right ) + 4 \, {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 8 \, c^{3} d^{6} - 56 \, a c^{2} d^{4} e^{2} + 175 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (13 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (40 \, c^{3} d^{5} e - 119 \, a c^{2} d^{3} e^{3} + 70 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{48 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + a^{2} c^{4} d^{4} e^{2}\right )}}, -\frac {105 \, {\left (a^{2} c^{2} d^{4} e^{3} - 2 \, a^{3} c d^{2} e^{5} + a^{4} e^{7} + {\left (c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (a c^{3} d^{5} e^{2} - 2 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} x\right )} \sqrt {-\frac {e}{c d}} \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 {-\frac {e}{c d}}}{2 \, {\left (c d e^{2} x^{2} + a d e^{2} + {\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, {\left (6 \, c^{3} d^{3} e^{3} x^{3} - 8 \, c^{3} d^{6} - 56 \, a c^{2} d^{4} e^{2} + 175 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (13 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 2 \, {\left (40 \, c^{3} d^{5} e - 119 \, a c^{2} d^{3} e^{3} + 70 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{24 \, {\left (c^{6} d^{6} x^{2} + 2 \, a c^{5} d^{5} e x + 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*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2
+ 2*(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(e/(c*d))*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 + 8*(c^2*d^3*e + a*c*d*e^3)*x + 4*(2*c^2*d^2*e*x + c^2*d^3 + a*c*d*e^2)*sqrt(c*d*e*x^2 + a*
d*e + (c*d^2 + a*e^2)*x)*sqrt(e/(c*d))) + 4*(6*c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*
e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*
d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2), -1/2
4*(105*(a^2*c^2*d^4*e^3 - 2*a^3*c*d^2*e^5 + a^4*e^7 + (c^4*d^6*e - 2*a*c^3*d^4*e^3 + a^2*c^2*d^2*e^5)*x^2 + 2*
(a*c^3*d^5*e^2 - 2*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*x)*sqrt(-e/(c*d))*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(-e/(c*d))/(c*d*e^2*x^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)) - 2*(6*
c^3*d^3*e^3*x^3 - 8*c^3*d^6 - 56*a*c^2*d^4*e^2 + 175*a^2*c*d^2*e^4 - 105*a^3*e^6 + 3*(13*c^3*d^4*e^2 - 7*a*c^2
*d^2*e^4)*x^2 - 2*(40*c^3*d^5*e - 119*a*c^2*d^3*e^3 + 70*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x))/(c^6*d^6*x^2 + 2*a*c^5*d^5*e*x + a^2*c^4*d^4*e^2)]

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giac [B]  time = 0.80, size = 1030, normalized size = 3.50 \begin {gather*} \frac {{\left ({\left ({\left (3 \, {\left (\frac {2 \, {\left (c^{7} d^{11} e^{8} - 4 \, a c^{6} d^{9} e^{10} + 6 \, a^{2} c^{5} d^{7} e^{12} - 4 \, a^{3} c^{4} d^{5} e^{14} + a^{4} c^{3} d^{3} e^{16}\right )} x}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}} + \frac {17 \, c^{7} d^{12} e^{7} - 75 \, a c^{6} d^{10} e^{9} + 130 \, a^{2} c^{5} d^{8} e^{11} - 110 \, a^{3} c^{4} d^{6} e^{13} + 45 \, a^{4} c^{3} d^{4} e^{15} - 7 \, a^{5} c^{2} d^{2} e^{17}}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}}\right )} x + \frac {4 \, {\left (c^{7} d^{13} e^{6} + 45 \, a c^{6} d^{11} e^{8} - 225 \, a^{2} c^{5} d^{9} e^{10} + 430 \, a^{3} c^{4} d^{7} e^{12} - 405 \, a^{4} c^{3} d^{5} e^{14} + 189 \, a^{5} c^{2} d^{3} e^{16} - 35 \, a^{6} c d e^{18}\right )}}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}}\right )} x - \frac {3 \, {\left (43 \, c^{7} d^{14} e^{5} - 305 \, a c^{6} d^{12} e^{7} + 825 \, a^{2} c^{5} d^{10} e^{9} - 1075 \, a^{3} c^{4} d^{8} e^{11} + 645 \, a^{4} c^{3} d^{6} e^{13} - 63 \, a^{5} c^{2} d^{4} e^{15} - 105 \, a^{6} c d^{2} e^{17} + 35 \, a^{7} e^{19}\right )}}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}}\right )} x - \frac {6 \, {\left (16 \, c^{7} d^{15} e^{4} - 85 \, a c^{6} d^{13} e^{6} + 145 \, a^{2} c^{5} d^{11} e^{8} - 15 \, a^{3} c^{4} d^{9} e^{10} - 250 \, a^{4} c^{3} d^{7} e^{12} + 329 \, a^{5} c^{2} d^{5} e^{14} - 175 \, a^{6} c d^{3} e^{16} + 35 \, a^{7} d e^{18}\right )}}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}}\right )} x - \frac {8 \, c^{7} d^{16} e^{3} + 24 \, a c^{6} d^{14} e^{5} - 351 \, a^{2} c^{5} d^{12} e^{7} + 1109 \, a^{3} c^{4} d^{10} e^{9} - 1686 \, a^{4} c^{3} d^{8} e^{11} + 1386 \, a^{5} c^{2} d^{6} e^{13} - 595 \, a^{6} c d^{4} e^{15} + 105 \, a^{7} d^{2} e^{17}}{c^{8} d^{12} e^{3} - 4 \, a c^{7} d^{10} e^{5} + 6 \, a^{2} c^{6} d^{8} e^{7} - 4 \, a^{3} c^{5} d^{6} e^{9} + a^{4} c^{4} d^{4} e^{11}}}{12 \, {\left (c d x^{2} e + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}} - \frac {35 \, {\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} \sqrt {c d} e^{\left (-\frac {1}{2}\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 + a d e + {\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt {c d} a e^{\frac {5}{2}} \right |}\right )}{8 \, c^{5} d^{5}} \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]

1/12*((((3*(2*(c^7*d^11*e^8 - 4*a*c^6*d^9*e^10 + 6*a^2*c^5*d^7*e^12 - 4*a^3*c^4*d^5*e^14 + a^4*c^3*d^3*e^16)*x
/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11) + (17*c^7*d^12*e
^7 - 75*a*c^6*d^10*e^9 + 130*a^2*c^5*d^8*e^11 - 110*a^3*c^4*d^6*e^13 + 45*a^4*c^3*d^4*e^15 - 7*a^5*c^2*d^2*e^1
7)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x + 4*(c^7*d^
13*e^6 + 45*a*c^6*d^11*e^8 - 225*a^2*c^5*d^9*e^10 + 430*a^3*c^4*d^7*e^12 - 405*a^4*c^3*d^5*e^14 + 189*a^5*c^2*
d^3*e^16 - 35*a^6*c*d*e^18)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4
*d^4*e^11))*x - 3*(43*c^7*d^14*e^5 - 305*a*c^6*d^12*e^7 + 825*a^2*c^5*d^10*e^9 - 1075*a^3*c^4*d^8*e^11 + 645*a
^4*c^3*d^6*e^13 - 63*a^5*c^2*d^4*e^15 - 105*a^6*c*d^2*e^17 + 35*a^7*e^19)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6
*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x - 6*(16*c^7*d^15*e^4 - 85*a*c^6*d^13*e^6 + 145*a^2
*c^5*d^11*e^8 - 15*a^3*c^4*d^9*e^10 - 250*a^4*c^3*d^7*e^12 + 329*a^5*c^2*d^5*e^14 - 175*a^6*c*d^3*e^16 + 35*a^
7*d*e^18)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))*x - (8
*c^7*d^16*e^3 + 24*a*c^6*d^14*e^5 - 351*a^2*c^5*d^12*e^7 + 1109*a^3*c^4*d^10*e^9 - 1686*a^4*c^3*d^8*e^11 + 138
6*a^5*c^2*d^6*e^13 - 595*a^6*c*d^4*e^15 + 105*a^7*d^2*e^17)/(c^8*d^12*e^3 - 4*a*c^7*d^10*e^5 + 6*a^2*c^6*d^8*e
^7 - 4*a^3*c^5*d^6*e^9 + a^4*c^4*d^4*e^11))/(c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2) - 35/8*(c^2*d^4*e^2
- 2*a*c*d^2*e^4 + a^2*e^6)*sqrt(c*d)*e^(-1/2)*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 + a*d*e + (c*d^2 + a*e^2)*x))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5)

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maple [B]  time = 0.14, size = 4008, normalized size = 13.63 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128/c*d^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+35/8*e^2/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)-35/8*e^2/c^2*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/
2)+35/16*e/c^2*d/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)+253/384*c*d^8/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e
*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-35/16*e^5/c^4/d^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+35/16*e^7/c^5/
d^5/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3+625/64*e^4/c^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+17/
4*e^4/c*x^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+35/16*e*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(1/2)+1/2*e^5*x^5/c/d/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+237/16*e^4/c^2*x^2/(c*d*e*x
^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-437/384*e^2*d^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d
^2)*x)^(3/2)*a-185/384*e^8/c^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4+35
/8*e^2*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x-165/16*e^5*d^5/(-a^2*e^4
+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2-35/16*e^3/c^3/d/(c*d*e*x^2+a*d*e+(a*e^2+
c*d^2)*x)^(1/2)*a-35/24*e^3/c*d*x^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-1165/192*e^6/c^4/d^2/(c*d*e*x^2+a*
d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3-1249/128*e^2/c^2*d^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-285/16*e^2/c*d^2
*x^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)-765/64*e/c*d^3*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)+637/384*
e^8/c^5/d^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4+253/48*e*c^2*d^9/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c
*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)-35/384*e^10/c^6/d^6/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^5-35/24*e^
14/c^4/d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^6+265/6*e^8/c*d^2/(-
a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^3-35/192*e^13/c^5/d^5/(-a^2*e^4+2
*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^6+77/32*e^11/c^4/d^3/(-a^2*e^4+2*a*c*d^2*e^2
-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^5+35/8*e^10/c^4/d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*
d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^4+65/64*e^5/c*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(
a*e^2+c*d^2)*x)^(3/2)*x*a^2-415/64*e^9/c^3/d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x
)^(3/2)*x*a^4-115/4*e^4*c*d^6/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a+7
7/4*e^12/c^3/d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^5+265/48*e^7/c
^2*d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a^3-35/384*e^14/c^6/d^6/(-a^2*
e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^7+427/384*e^12/c^5/d^4/(-a^2*e^4+2*a*c*d^
2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^6-35/4*e^6/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x*a^2-415/8*e^10/c^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a
*e^2+c*d^2)*x)^(1/2)*x*a^4+65/8*e^6*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(
1/2)*x*a^2-115/32*e^3*d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x*a-165/128
*e^4/c*d^4/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2-35/8*e^7/c^3/d/(-a^2*e
^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^3-35/8*e^5/c^2*d/(-a^2*e^4+2*a*c*d^2*e^2-c
^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^2+35/12*e^5/c^2/d*x^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2
)*a+35/16*e^3/c*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a+45/32*e^5/c^3/d
*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2-7/4*e^6/c^2/d^2*x^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-3
5/4*e^4/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-3/4*e^3/c^2*d*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a-437/48*e^3*c*d^7/(-a^2*e^4+2*a*c*d^2*e^2-c
^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a+35/4*e^4/c^3/d^2*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)
*a-261/128*e^10/c^4/d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^5+1255/384*
e^6/c^2*d^2/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3-147/16*e^6/c^3/d^2*x^
2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+35/64*e^9/c^5/d^5*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^4-
7/4*e^7/c^4/d^3*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3-35/48*e^15/c^5/d^5/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d
^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^7+427/48*e^13/c^4/d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*
e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^6-261/16*e^11/c^3/d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(
a*e^2+c*d^2)*x)^(1/2)*a^5-35/24*e^7/c^3/d^3*x^3/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^2+35/16*e^11/c^5/d^5
/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^5+35/16*e^9/c^4/d^3/(-a^2*e^4+2*a*
c*d^2*e^2-c^2*d^4)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^4-35/8*e^6/c^4/d^4*x/(c*d*e*x^2+a*d*e+(a*e^2+c*d^
2)*x)^(1/2)*a^2+35/8*e^6/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^2-185/48*e^9/c^2*d/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2
)*x)^(1/2)*a^4+1255/48*e^7/c*d^3/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*a^
3+35/16*e^8/c^4/d^4*x^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*a^3+253/24*e^2*c^2*d^8/(-a^2*e^4+2*a*c*d^2*e^2
-c^2*d^4)^2/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)*x+253/192*e*c*d^7/(-a^2*e^4+2*a*c*d^2*e^2-c^2*d^4)/(c*d*e*
x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2)*x

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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(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 \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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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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