3.2009 \(\int \frac {(d+e x)^{13/2}}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx\)

Optimal. Leaf size=210 \[ -\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}+\frac {9 e \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}+\frac {3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2} \]

[Out]

3*e*(-a*e^2+c*d^2)^2*(e*x+d)^(3/2)/c^4/d^4+9/5*e*(-a*e^2+c*d^2)*(e*x+d)^(5/2)/c^3/d^3+9/7*e*(e*x+d)^(7/2)/c^2/
d^2-(e*x+d)^(9/2)/c/d/(c*d*x+a*e)-9*e*(-a*e^2+c*d^2)^(7/2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)^(1/2)/(-a*e^2+c*d^2
)^(1/2))/c^(11/2)/d^(11/2)+9*e*(-a*e^2+c*d^2)^3*(e*x+d)^(1/2)/c^5/d^5

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Rubi [A]  time = 0.18, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {626, 47, 50, 63, 208} \[ \frac {9 e (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 c^3 d^3}+\frac {3 e (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{c^4 d^4}+\frac {9 e \sqrt {d+e x} \left (c d^2-a e^2\right )^3}{c^5 d^5}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(9*e*(c*d^2 - a*e^2)^3*Sqrt[d + e*x])/(c^5*d^5) + (3*e*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2))/(c^4*d^4) + (9*e*(c*
d^2 - a*e^2)*(d + e*x)^(5/2))/(5*c^3*d^3) + (9*e*(d + e*x)^(7/2))/(7*c^2*d^2) - (d + e*x)^(9/2)/(c*d*(a*e + c*
d*x)) - (9*e*(c*d^2 - a*e^2)^(7/2)*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c^(11/2)*d^(
11/2))

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 626

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

Rubi steps

\begin {align*} \int \frac {(d+e x)^{13/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx &=\int \frac {(d+e x)^{9/2}}{(a e+c d x)^2} \, dx\\ &=-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{a e+c d x} \, dx}{2 c d}\\ &=\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )\right ) \int \frac {(d+e x)^{5/2}}{a e+c d x} \, dx}{2 c^2 d^2}\\ &=\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^2\right ) \int \frac {(d+e x)^{3/2}}{a e+c d x} \, dx}{2 c^3 d^3}\\ &=\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^3\right ) \int \frac {\sqrt {d+e x}}{a e+c d x} \, dx}{2 c^4 d^4}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 e \left (c d^2-a e^2\right )^4\right ) \int \frac {1}{(a e+c d x) \sqrt {d+e x}} \, dx}{2 c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}+\frac {\left (9 \left (c d^2-a e^2\right )^4\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {c d^2}{e}+a e+\frac {c d x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{c^5 d^5}\\ &=\frac {9 e \left (c d^2-a e^2\right )^3 \sqrt {d+e x}}{c^5 d^5}+\frac {3 e \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{c^4 d^4}+\frac {9 e \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 c^3 d^3}+\frac {9 e (d+e x)^{7/2}}{7 c^2 d^2}-\frac {(d+e x)^{9/2}}{c d (a e+c d x)}-\frac {9 e \left (c d^2-a e^2\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d} \sqrt {d+e x}}{\sqrt {c d^2-a e^2}}\right )}{c^{11/2} d^{11/2}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 59, normalized size = 0.28 \[ \frac {2 e (d+e x)^{11/2} \, _2F_1\left (2,\frac {11}{2};\frac {13}{2};-\frac {c d (d+e x)}{a e^2-c d^2}\right )}{11 \left (a e^2-c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^(13/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

(2*e*(d + e*x)^(11/2)*Hypergeometric2F1[2, 11/2, 13/2, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/(11*(-(c*d^2) +
a*e^2)^2)

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fricas [B]  time = 1.23, size = 788, normalized size = 3.75 \[ \left [\frac {315 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {\frac {c d^{2} - a e^{2}}{c d}} \log \left (\frac {c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt {e x + d} c d \sqrt {\frac {c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, {\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \, {\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt {e x + d}}{70 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}, -\frac {315 \, {\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} + {\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \sqrt {-\frac {c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac {\sqrt {e x + d} c d \sqrt {-\frac {c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - {\left (10 \, c^{4} d^{4} e^{4} x^{4} - 35 \, c^{4} d^{8} + 528 \, a c^{3} d^{6} e^{2} - 1218 \, a^{2} c^{2} d^{4} e^{4} + 1050 \, a^{3} c d^{2} e^{6} - 315 \, a^{4} e^{8} + 2 \, {\left (29 \, c^{4} d^{5} e^{3} - 9 \, a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \, {\left (26 \, c^{4} d^{6} e^{2} - 23 \, a c^{3} d^{4} e^{4} + 7 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (194 \, c^{4} d^{7} e - 426 \, a c^{3} d^{5} e^{3} + 357 \, a^{2} c^{2} d^{3} e^{5} - 105 \, a^{3} c d e^{7}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="fricas")

[Out]

[1/70*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4 + 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a
^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt((c*d^2 - a*e^2)/(c*d))*log((c*d*e*x + 2*c*d^2 - a*e^2 - 2*sqrt(e*x + d)*
c*d*sqrt((c*d^2 - a*e^2)/(c*d)))/(c*d*x + a*e)) + 2*(10*c^4*d^4*e^4*x^4 - 35*c^4*d^8 + 528*a*c^3*d^6*e^2 - 121
8*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e^3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^
6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e - 426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5
- 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*e), -1/35*(315*(a*c^3*d^6*e^2 - 3*a^2*c^2*d^4*e^4
+ 3*a^3*c*d^2*e^6 - a^4*e^8 + (c^4*d^7*e - 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x)*sqrt(-(c*d^2
- a*e^2)/(c*d))*arctan(-sqrt(e*x + d)*c*d*sqrt(-(c*d^2 - a*e^2)/(c*d))/(c*d^2 - a*e^2)) - (10*c^4*d^4*e^4*x^4
- 35*c^4*d^8 + 528*a*c^3*d^6*e^2 - 1218*a^2*c^2*d^4*e^4 + 1050*a^3*c*d^2*e^6 - 315*a^4*e^8 + 2*(29*c^4*d^5*e^
3 - 9*a*c^3*d^3*e^5)*x^3 + 6*(26*c^4*d^6*e^2 - 23*a*c^3*d^4*e^4 + 7*a^2*c^2*d^2*e^6)*x^2 + 2*(194*c^4*d^7*e -
426*a*c^3*d^5*e^3 + 357*a^2*c^2*d^3*e^5 - 105*a^3*c*d*e^7)*x)*sqrt(e*x + d))/(c^6*d^6*x + a*c^5*d^5*e)]

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.06, size = 628, normalized size = 2.99 \[ \frac {9 a^{4} e^{9} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{5} d^{5}}-\frac {36 a^{3} e^{7} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{4} d^{3}}+\frac {54 a^{2} e^{5} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{3} d}-\frac {36 a d \,e^{3} \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c^{2}}+\frac {9 d^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c d}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}}\right )}{\sqrt {\left (a \,e^{2}-c \,d^{2}\right ) c d}\, c}-\frac {\sqrt {e x +d}\, a^{4} e^{9}}{\left (c d e x +a \,e^{2}\right ) c^{5} d^{5}}+\frac {4 \sqrt {e x +d}\, a^{3} e^{7}}{\left (c d e x +a \,e^{2}\right ) c^{4} d^{3}}-\frac {6 \sqrt {e x +d}\, a^{2} e^{5}}{\left (c d e x +a \,e^{2}\right ) c^{3} d}+\frac {4 \sqrt {e x +d}\, a d \,e^{3}}{\left (c d e x +a \,e^{2}\right ) c^{2}}-\frac {\sqrt {e x +d}\, d^{3} e}{\left (c d e x +a \,e^{2}\right ) c}-\frac {8 \sqrt {e x +d}\, a^{3} e^{7}}{c^{5} d^{5}}+\frac {24 \sqrt {e x +d}\, a^{2} e^{5}}{c^{4} d^{3}}-\frac {24 \sqrt {e x +d}\, a \,e^{3}}{c^{3} d}+\frac {8 \sqrt {e x +d}\, d e}{c^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} a^{2} e^{5}}{c^{4} d^{4}}-\frac {4 \left (e x +d \right )^{\frac {3}{2}} a \,e^{3}}{c^{3} d^{2}}+\frac {2 \left (e x +d \right )^{\frac {3}{2}} e}{c^{2}}-\frac {4 \left (e x +d \right )^{\frac {5}{2}} a \,e^{3}}{5 c^{3} d^{3}}+\frac {4 \left (e x +d \right )^{\frac {5}{2}} e}{5 c^{2} d}+\frac {2 \left (e x +d \right )^{\frac {7}{2}} e}{7 c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(13/2)/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^2,x)

[Out]

2/7*e*(e*x+d)^(7/2)/c^2/d^2-4/5/c^3/d^3*(e*x+d)^(5/2)*a*e^3+4/5*e/c^2/d*(e*x+d)^(5/2)+2/c^4/d^4*(e*x+d)^(3/2)*
a^2*e^5-4/c^3/d^2*(e*x+d)^(3/2)*a*e^3+2*e/c^2*(e*x+d)^(3/2)-8/c^5/d^5*a^3*e^7*(e*x+d)^(1/2)+24/c^4/d^3*a^2*e^5
*(e*x+d)^(1/2)-24/c^3/d*a*e^3*(e*x+d)^(1/2)+8*e/c^2*d*(e*x+d)^(1/2)-1/c^5/d^5*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^
4*e^9+4/c^4/d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^3*e^7-6/c^3/d*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)*a^2*e^5+4/c^2*d*(e
*x+d)^(1/2)/(c*d*e*x+a*e^2)*a*e^3-e/c*d^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+9/c^5/d^5/((a*e^2-c*d^2)*c*d)^(1/2)*ar
ctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)*a^4*e^9-36/c^4/d^3/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^
(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)*a^3*e^7+54/c^3/d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c
*d^2)*c*d)^(1/2)*c*d)*a^2*e^5-36/c^2*d/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2
)*c*d)*a*e^3+9*e/c*d^3/((a*e^2-c*d^2)*c*d)^(1/2)*arctan((e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2)*c*d)

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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)^(13/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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 positive or negative?

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mupad [B]  time = 0.70, size = 443, normalized size = 2.11 \[ \frac {2\,e\,{\left (d+e\,x\right )}^{7/2}}{7\,c^2\,d^2}-\left (\frac {\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{c^6\,d^6}\right )}{c^2\,d^2}+\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}{c^6\,d^6}\right )\,\sqrt {d+e\,x}-\frac {\sqrt {d+e\,x}\,\left (a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e\right )}{c^6\,d^6\,\left (d+e\,x\right )-c^6\,d^7+a\,c^5\,d^5\,e^2}-\left (\frac {2\,e\,{\left (a\,e^2-c\,d^2\right )}^2}{3\,c^4\,d^4}-\frac {2\,e\,{\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )}^2}{3\,c^6\,d^6}\right )\,{\left (d+e\,x\right )}^{3/2}+\frac {2\,e\,\left (2\,c^2\,d^3-2\,a\,c\,d\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,c^4\,d^4}+\frac {9\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,\sqrt {d}\,e\,{\left (a\,e^2-c\,d^2\right )}^{7/2}\,\sqrt {d+e\,x}}{a^4\,e^9-4\,a^3\,c\,d^2\,e^7+6\,a^2\,c^2\,d^4\,e^5-4\,a\,c^3\,d^6\,e^3+c^4\,d^8\,e}\right )\,{\left (a\,e^2-c\,d^2\right )}^{7/2}}{c^{11/2}\,d^{11/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(13/2)/(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^2,x)

[Out]

(2*e*(d + e*x)^(7/2))/(7*c^2*d^2) - (((2*c^2*d^3 - 2*a*c*d*e^2)*((2*e*(a*e^2 - c*d^2)^2)/(c^4*d^4) - (2*e*(2*c
^2*d^3 - 2*a*c*d*e^2)^2)/(c^6*d^6)))/(c^2*d^2) + (2*e*(a*e^2 - c*d^2)^2*(2*c^2*d^3 - 2*a*c*d*e^2))/(c^6*d^6))*
(d + e*x)^(1/2) - ((d + e*x)^(1/2)*(a^4*e^9 + c^4*d^8*e - 4*a*c^3*d^6*e^3 - 4*a^3*c*d^2*e^7 + 6*a^2*c^2*d^4*e^
5))/(c^6*d^6*(d + e*x) - c^6*d^7 + a*c^5*d^5*e^2) - ((2*e*(a*e^2 - c*d^2)^2)/(3*c^4*d^4) - (2*e*(2*c^2*d^3 - 2
*a*c*d*e^2)^2)/(3*c^6*d^6))*(d + e*x)^(3/2) + (2*e*(2*c^2*d^3 - 2*a*c*d*e^2)*(d + e*x)^(5/2))/(5*c^4*d^4) + (9
*e*atan((c^(1/2)*d^(1/2)*e*(a*e^2 - c*d^2)^(7/2)*(d + e*x)^(1/2))/(a^4*e^9 + c^4*d^8*e - 4*a*c^3*d^6*e^3 - 4*a
^3*c*d^2*e^7 + 6*a^2*c^2*d^4*e^5))*(a*e^2 - c*d^2)^(7/2))/(c^(11/2)*d^(11/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(13/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

Timed out

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