3.21.72 \(\int \frac {1}{(d+e x)^{7/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\) [2072]

Optimal. Leaf size=393 \[ \frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3}{64 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 \left (c d^2-a e^2\right )^{11/2}} \]

[Out]

-315/64*c^4*d^4*arctan(e^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(-a*e^2+c*d^2)^(1/2)/(e*x+d)^(1/2))*e^(
1/2)/(-a*e^2+c*d^2)^(11/2)+1/4/(-a*e^2+c*d^2)/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/8*c*d/(-
a*e^2+c*d^2)^2/(e*x+d)^(5/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+21/32*c^2*d^2/(-a*e^2+c*d^2)^3/(e*x+d)^(3
/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+105/64*c^3*d^3/(-a*e^2+c*d^2)^4/(e*x+d)^(1/2)/(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2)-315/64*c^4*d^4*(e*x+d)^(1/2)/(-a*e^2+c*d^2)^5/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {686, 680, 674, 211} \begin {gather*} -\frac {315 c^4 d^4 \sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt {d+e x} \sqrt {c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{11/2}}-\frac {315 c^4 d^4 \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {105 c^3 d^3}{64 \sqrt {d+e x} \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {21 c^2 d^2}{32 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {3 c d}{8 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {1}{4 (d+e x)^{7/2} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

1/(4*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (3*c*d)/(8*(c*d^2 - a*e^2)
^2*(d + e*x)^(5/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (21*c^2*d^2)/(32*(c*d^2 - a*e^2)^3*(d + e*x)
^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (105*c^3*d^3)/(64*(c*d^2 - a*e^2)^4*Sqrt[d + e*x]*Sqrt[a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4*Sqrt[d + e*x])/(64*(c*d^2 - a*e^2)^5*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2]) - (315*c^4*d^4*Sqrt[e]*ArcTan[(Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(Sqrt[c*d^2 - a*e^2]*Sqrt[d + e*x])])/(64*(c*d^2 - a*e^2)^(11/2))

Rule 211

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

Rule 674

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2*e, Subst[Int[1/(
2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + b*x + c*x^2]/Sqrt[d + e*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]

Rule 680

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(2*c*d - b*e)*(d + e
*x)^m*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(2*c*d - b*e)*((m + 2*p + 2)/((p + 1)*(
b^2 - 4*a*c))), Int[(d + e*x)^(m - 1)*(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] && LtQ[0, m, 1] && IntegerQ[2*p]

Rule 686

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-e)*(d + e*x)^m*((a
+ b*x + c*x^2)^(p + 1)/((m + p + 1)*(2*c*d - b*e))), x] + Dist[c*((m + 2*p + 2)/((m + p + 1)*(2*c*d - b*e))),
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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, 0] && NeQ[m + p + 1, 0] && IntegerQ[2*p]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{7/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(9 c d) \int \frac {1}{(d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{8 \left (c d^2-a e^2\right )}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (21 c^2 d^2\right ) \int \frac {1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{16 \left (c d^2-a e^2\right )^2}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (105 c^3 d^3\right ) \int \frac {1}{\sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{64 \left (c d^2-a e^2\right )^3}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3}{64 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (315 c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{128 \left (c d^2-a e^2\right )^4}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3}{64 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (315 c^4 d^4 e\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{128 \left (c d^2-a e^2\right )^5}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3}{64 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {\left (315 c^4 d^4 e^2\right ) \text {Subst}\left (\int \frac {1}{2 c d^2 e-e \left (c d^2+a e^2\right )+e^2 x^2} \, dx,x,\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}}\right )}{64 \left (c d^2-a e^2\right )^5}\\ &=\frac {1}{4 \left (c d^2-a e^2\right ) (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3 c d}{8 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {21 c^2 d^2}{32 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {105 c^3 d^3}{64 \left (c d^2-a e^2\right )^4 \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {d+e x}}{64 \left (c d^2-a e^2\right )^5 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {315 c^4 d^4 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c d^2-a e^2} \sqrt {d+e x}}\right )}{64 \left (c d^2-a e^2\right )^{11/2}}\\ \end {align*}

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Mathematica [A]
time = 1.13, size = 282, normalized size = 0.72 \begin {gather*} \frac {-\sqrt {c d^2-a e^2} \left (-16 a^4 e^8+8 a^3 c d e^6 (11 d+3 e x)-6 a^2 c^2 d^2 e^4 \left (35 d^2+26 d e x+7 e^2 x^2\right )+a c^3 d^3 e^2 \left (325 d^3+555 d^2 e x+399 d e^2 x^2+105 e^3 x^3\right )+c^4 d^4 \left (128 d^4+837 d^3 e x+1533 d^2 e^2 x^2+1155 d e^3 x^3+315 e^4 x^4\right )\right )-315 c^4 d^4 \sqrt {e} \sqrt {a e+c d x} (d+e x)^4 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c d^2-a e^2}}\right )}{64 \left (c d^2-a e^2\right )^{11/2} (d+e x)^{7/2} \sqrt {(a e+c d x) (d+e x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(-(Sqrt[c*d^2 - a*e^2]*(-16*a^4*e^8 + 8*a^3*c*d*e^6*(11*d + 3*e*x) - 6*a^2*c^2*d^2*e^4*(35*d^2 + 26*d*e*x + 7*
e^2*x^2) + a*c^3*d^3*e^2*(325*d^3 + 555*d^2*e*x + 399*d*e^2*x^2 + 105*e^3*x^3) + c^4*d^4*(128*d^4 + 837*d^3*e*
x + 1533*d^2*e^2*x^2 + 1155*d*e^3*x^3 + 315*e^4*x^4))) - 315*c^4*d^4*Sqrt[e]*Sqrt[a*e + c*d*x]*(d + e*x)^4*Arc
Tan[(Sqrt[e]*Sqrt[a*e + c*d*x])/Sqrt[c*d^2 - a*e^2]])/(64*(c*d^2 - a*e^2)^(11/2)*(d + e*x)^(7/2)*Sqrt[(a*e + c
*d*x)*(d + e*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(756\) vs. \(2(349)=698\).
time = 0.96, size = 757, normalized size = 1.93

method	result	size
default	\(-\frac {\sqrt {\left (c d x +a e \right ) \left (e x +d \right )}\, \left (315 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{4} e^{5} x^{4}+1260 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{5} e^{4} x^{3}+1890 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{6} e^{3} x^{2}-315 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{4} e^{4} x^{4}+1260 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{7} e^{2} x -105 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{3} e^{5} x^{3}-1155 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{5} e^{3} x^{3}+315 \sqrt {c d x +a e}\, \arctanh \left (\frac {e \sqrt {c d x +a e}}{\sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\right ) c^{4} d^{8} e +42 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{2} e^{6} x^{2}-399 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{4} e^{4} x^{2}-1533 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{6} e^{2} x^{2}-24 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c d \,e^{7} x +156 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{3} e^{5} x -555 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{5} e^{3} x -837 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{7} e x +16 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{4} e^{8}-88 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{3} c \,d^{2} e^{6}+210 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a^{2} c^{2} d^{4} e^{4}-325 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, a \,c^{3} d^{6} e^{2}-128 \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}\, c^{4} d^{8}\right )}{64 \left (e x +d \right )^{\frac {9}{2}} \left (c d x +a e \right ) \left (e^{2} a -c \,d^{2}\right )^{5} \sqrt {\left (e^{2} a -c \,d^{2}\right ) e}}\)	\(757\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/64/(e*x+d)^(9/2)*((c*d*x+a*e)*(e*x+d))^(1/2)*(315*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d
^2)*e)^(1/2))*c^4*d^4*e^5*x^4+1260*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*
d^5*e^4*x^3+1890*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^6*e^3*x^2-315*((
a*e^2-c*d^2)*e)^(1/2)*c^4*d^4*e^4*x^4+1260*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/
2))*c^4*d^7*e^2*x-105*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^3*e^5*x^3-1155*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^5*e^3*x^3+3
15*(c*d*x+a*e)^(1/2)*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c^4*d^8*e+42*((a*e^2-c*d^2)*e)^(1/2)
*a^2*c^2*d^2*e^6*x^2-399*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^4*e^4*x^2-1533*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^6*e^2*x^
2-24*((a*e^2-c*d^2)*e)^(1/2)*a^3*c*d*e^7*x+156*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^3*e^5*x-555*((a*e^2-c*d^2)*e)
^(1/2)*a*c^3*d^5*e^3*x-837*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^7*e*x+16*((a*e^2-c*d^2)*e)^(1/2)*a^4*e^8-88*((a*e^2-c
*d^2)*e)^(1/2)*a^3*c*d^2*e^6+210*((a*e^2-c*d^2)*e)^(1/2)*a^2*c^2*d^4*e^4-325*((a*e^2-c*d^2)*e)^(1/2)*a*c^3*d^6
*e^2-128*((a*e^2-c*d^2)*e)^(1/2)*c^4*d^8)/(c*d*x+a*e)/(a*e^2-c*d^2)^5/((a*e^2-c*d^2)*e)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((c*d*x^2*e + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(x*e + d)^(7/2)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1023 vs. \(2 (352) = 704\).
time = 4.81, size = 2084, normalized size = 5.30 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")

[Out]

[1/128*(315*(c^5*d^10*x + a*c^4*d^4*x^5*e^6 + (c^5*d^5*x^6 + 5*a*c^4*d^5*x^4)*e^5 + 5*(c^5*d^6*x^5 + 2*a*c^4*d
^6*x^3)*e^4 + 10*(c^5*d^7*x^4 + a*c^4*d^7*x^2)*e^3 + 5*(2*c^5*d^8*x^3 + a*c^4*d^8*x)*e^2 + (5*c^5*d^9*x^2 + a*
c^4*d^9)*e)*sqrt(-e/(c*d^2 - a*e^2))*log((c*d^3 - 2*a*x*e^3 + 2*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*(c
*d^2 - a*e^2)*sqrt(x*e + d)*sqrt(-e/(c*d^2 - a*e^2)) - (c*d*x^2 + 2*a*d)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)) - 2*(
837*c^4*d^7*x*e + 128*c^4*d^8 + 24*a^3*c*d*x*e^7 - 16*a^4*e^8 - 2*(21*a^2*c^2*d^2*x^2 - 44*a^3*c*d^2)*e^6 + 3*
(35*a*c^3*d^3*x^3 - 52*a^2*c^2*d^3*x)*e^5 + 21*(15*c^4*d^4*x^4 + 19*a*c^3*d^4*x^2 - 10*a^2*c^2*d^4)*e^4 + 15*(
77*c^4*d^5*x^3 + 37*a*c^3*d^5*x)*e^3 + (1533*c^4*d^6*x^2 + 325*a*c^3*d^6)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x
^2 + a*d)*e)*sqrt(x*e + d))/(10*c^6*d^14*x^3*e^2 + c^6*d^16*x - a^6*x^5*e^16 - 10*a^6*d^2*x^3*e^14 - (a^5*c*d*
x^6 + 5*a^6*d*x^4)*e^15 + 5*(a^4*c^2*d^3*x^6 + 3*a^5*c*d^3*x^4 - 2*a^6*d^3*x^2)*e^13 + 5*(3*a^4*c^2*d^4*x^5 +
8*a^5*c*d^4*x^3 - a^6*d^4*x)*e^12 - (10*a^3*c^3*d^5*x^6 - 45*a^5*c*d^5*x^2 + a^6*d^5)*e^11 - 2*(20*a^3*c^3*d^6
*x^5 + 25*a^4*c^2*d^6*x^3 - 12*a^5*c*d^6*x)*e^10 + 5*(2*a^2*c^4*d^7*x^6 - 10*a^3*c^3*d^7*x^4 - 15*a^4*c^2*d^7*
x^2 + a^5*c*d^7)*e^9 + 45*(a^2*c^4*d^8*x^5 - a^4*c^2*d^8*x)*e^8 - 5*(a*c^5*d^9*x^6 - 15*a^2*c^4*d^9*x^4 - 10*a
^3*c^3*d^9*x^2 + 2*a^4*c^2*d^9)*e^7 - 2*(12*a*c^5*d^10*x^5 - 25*a^2*c^4*d^10*x^3 - 20*a^3*c^3*d^10*x)*e^6 + (c
^6*d^11*x^6 - 45*a*c^5*d^11*x^4 + 10*a^3*c^3*d^11)*e^5 + 5*(c^6*d^12*x^5 - 8*a*c^5*d^12*x^3 - 3*a^2*c^4*d^12*x
)*e^4 + 5*(2*c^6*d^13*x^4 - 3*a*c^5*d^13*x^2 - a^2*c^4*d^13)*e^3 + (5*c^6*d^15*x^2 + a*c^5*d^15)*e), -1/64*(31
5*(c^5*d^10*x + a*c^4*d^4*x^5*e^6 + (c^5*d^5*x^6 + 5*a*c^4*d^5*x^4)*e^5 + 5*(c^5*d^6*x^5 + 2*a*c^4*d^6*x^3)*e^
4 + 10*(c^5*d^7*x^4 + a*c^4*d^7*x^2)*e^3 + 5*(2*c^5*d^8*x^3 + a*c^4*d^8*x)*e^2 + (5*c^5*d^9*x^2 + a*c^4*d^9)*e
)*arctan(-sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(c*d^2 - a*e^2)*sqrt(x*e + d)*e^(1/2)/(c*d^2*x*e + a
*x*e^3 + (c*d*x^2 + a*d)*e^2))*e^(1/2)/sqrt(c*d^2 - a*e^2) + (837*c^4*d^7*x*e + 128*c^4*d^8 + 24*a^3*c*d*x*e^7
- 16*a^4*e^8 - 2*(21*a^2*c^2*d^2*x^2 - 44*a^3*c*d^2)*e^6 + 3*(35*a*c^3*d^3*x^3 - 52*a^2*c^2*d^3*x)*e^5 + 21*(
15*c^4*d^4*x^4 + 19*a*c^3*d^4*x^2 - 10*a^2*c^2*d^4)*e^4 + 15*(77*c^4*d^5*x^3 + 37*a*c^3*d^5*x)*e^3 + (1533*c^4
*d^6*x^2 + 325*a*c^3*d^6)*e^2)*sqrt(c*d^2*x + a*x*e^2 + (c*d*x^2 + a*d)*e)*sqrt(x*e + d))/(10*c^6*d^14*x^3*e^2
+ c^6*d^16*x - a^6*x^5*e^16 - 10*a^6*d^2*x^3*e^14 - (a^5*c*d*x^6 + 5*a^6*d*x^4)*e^15 + 5*(a^4*c^2*d^3*x^6 + 3
*a^5*c*d^3*x^4 - 2*a^6*d^3*x^2)*e^13 + 5*(3*a^4*c^2*d^4*x^5 + 8*a^5*c*d^4*x^3 - a^6*d^4*x)*e^12 - (10*a^3*c^3*
d^5*x^6 - 45*a^5*c*d^5*x^2 + a^6*d^5)*e^11 - 2*(20*a^3*c^3*d^6*x^5 + 25*a^4*c^2*d^6*x^3 - 12*a^5*c*d^6*x)*e^10
+ 5*(2*a^2*c^4*d^7*x^6 - 10*a^3*c^3*d^7*x^4 - 15*a^4*c^2*d^7*x^2 + a^5*c*d^7)*e^9 + 45*(a^2*c^4*d^8*x^5 - a^4
*c^2*d^8*x)*e^8 - 5*(a*c^5*d^9*x^6 - 15*a^2*c^4*d^9*x^4 - 10*a^3*c^3*d^9*x^2 + 2*a^4*c^2*d^9)*e^7 - 2*(12*a*c^
5*d^10*x^5 - 25*a^2*c^4*d^10*x^3 - 20*a^3*c^3*d^10*x)*e^6 + (c^6*d^11*x^6 - 45*a*c^5*d^11*x^4 + 10*a^3*c^3*d^1
1)*e^5 + 5*(c^6*d^12*x^5 - 8*a*c^5*d^12*x^3 - 3*a^2*c^4*d^12*x)*e^4 + 5*(2*c^6*d^13*x^4 - 3*a*c^5*d^13*x^2 - a
^2*c^4*d^13)*e^3 + (5*c^6*d^15*x^2 + a*c^5*d^15)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [A]
time = 2.04, size = 691, normalized size = 1.76 \begin {gather*} -\frac {315 \, c^{4} d^{4} \arctan \left (\frac {\sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}}{\sqrt {c d^{2} e - a e^{3}}}\right ) e}{64 \, {\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^{2} e - a e^{3}}} - \frac {2 \, c^{4} d^{4} e}{{\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 {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}}} - \frac {{\left (325 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} c^{7} d^{10} e^{4} - 975 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a c^{6} d^{8} e^{6} + 765 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} c^{6} d^{8} e^{3} + 975 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{2} c^{5} d^{6} e^{8} - 1530 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a c^{5} d^{6} e^{5} + 643 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} c^{5} d^{6} e^{2} - 325 \, \sqrt {{\left (x e + d\right )} c d e - c d^{2} e + a e^{3}} a^{3} c^{4} d^{4} e^{10} + 765 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} c^{4} d^{4} e^{7} - 643 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a c^{4} d^{4} e^{4} + 187 \, {\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} c^{4} d^{4} e\right )} e^{\left (-4\right )}}{64 \, {\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 )} {\left (x e + d\right )}^{4} c^{4} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")

[Out]

-315/64*c^4*d^4*arctan(sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)/sqrt(c*d^2*e - a*e^3))*e/((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^2*e - a*e^3)) - 2*c^4*
d^4*e/((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)*sqr
t((x*e + d)*c*d*e - c*d^2*e + a*e^3)) - 1/64*(325*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*c^7*d^10*e^4 - 975*s
qrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a*c^6*d^8*e^6 + 765*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*c^6*d^8*e
^3 + 975*sqrt((x*e + d)*c*d*e - c*d^2*e + a*e^3)*a^2*c^5*d^6*e^8 - 1530*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3
/2)*a*c^5*d^6*e^5 + 643*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*c^5*d^6*e^2 - 325*sqrt((x*e + d)*c*d*e - c*d
^2*e + a*e^3)*a^3*c^4*d^4*e^10 + 765*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^2*c^4*d^4*e^7 - 643*((x*e + d
)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*c^4*d^4*e^4 + 187*((x*e + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*c^4*d^4*e)*e^(-
4)/((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)*(x*e +
d)^4*c^4*d^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((d + e*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)

[Out]

int(1/((d + e*x)^(7/2)*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)), x)

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