3.17.49 \(\int \frac {1}{(d+e x)^4 (a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}} \, dx\)
Optimal. Leaf size=301 \[ -\frac {256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {64 c^3 d^3}{63 (d+e x) \left (c d^2-a e^2\right )^4 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {32 c^2 d^2}{63 (d+e x)^2 \left (c d^2-a e^2\right )^3 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {20 c d}{63 (d+e x)^3 \left (c d^2-a e^2\right )^2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {2}{9 (d+e x)^4 \left (c d^2-a e^2\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.16, antiderivative size = 301, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used =
{658, 613} \begin {gather*} -\frac {256 c^4 d^4 \left (a e^2+c d^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac {64 c^3 d^3}{63 (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 {32 c^2 d^2}{63 (d+e x)^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 {20 c d}{63 (d+e x)^3 \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 {2}{9 (d+e x)^4 \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)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
2/(9*(c*d^2 - a*e^2)*(d + e*x)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (20*c*d)/(63*(c*d^2 - a*e^2)^2
*(d + e*x)^3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (32*c^2*d^2)/(63*(c*d^2 - a*e^2)^3*(d + e*x)^2*Sqr
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) + (64*c^3*d^3)/(63*(c*d^2 - a*e^2)^4*(d + e*x)*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2]) - (256*c^4*d^4*(c*d^2 + a*e^2 + 2*c*d*e*x))/(63*(c*d^2 - a*e^2)^6*Sqrt[a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2])
Rule 613
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 658
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*Simplify[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, m, p}, x] && NeQ[b^2 - 4*a*c
, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2], 0]
Rubi steps
\begin {align*} \int \frac {1}{(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 &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {(10 c d) \int \frac {1}{(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}} \, dx}{9 \left (c d^2-a e^2\right )}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (80 c^2 d^2\right ) \int \frac {1}{(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}} \, dx}{63 \left (c d^2-a e^2\right )^2}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (32 c^3 d^3\right ) \int \frac {1}{(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}{21 \left (c d^2-a e^2\right )^3}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (128 c^4 d^4\right ) \int \frac {1}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{63 \left (c d^2-a e^2\right )^4}\\ &=\frac {2}{9 \left (c d^2-a e^2\right ) (d+e x)^4 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {20 c d}{63 \left (c d^2-a e^2\right )^2 (d+e x)^3 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {32 c^2 d^2}{63 \left (c d^2-a e^2\right )^3 (d+e x)^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {64 c^3 d^3}{63 \left (c d^2-a e^2\right )^4 (d+e x) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}-\frac {256 c^4 d^4 \left (c d^2+a e^2+2 c d e x\right )}{63 \left (c d^2-a e^2\right )^6 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 258, normalized size = 0.86 \begin {gather*} -\frac {2 \left (7 a^5 e^{10}-5 a^4 c d e^8 (9 d+2 e x)+2 a^3 c^2 d^2 e^6 \left (63 d^2+36 d e x+8 e^2 x^2\right )-2 a^2 c^3 d^3 e^4 \left (105 d^3+126 d^2 e x+72 d e^2 x^2+16 e^3 x^3\right )+a c^4 d^4 e^2 \left (315 d^4+840 d^3 e x+1008 d^2 e^2 x^2+576 d e^3 x^3+128 e^4 x^4\right )+c^5 d^5 \left (63 d^5+630 d^4 e x+1680 d^3 e^2 x^2+2016 d^2 e^3 x^3+1152 d e^4 x^4+256 e^5 x^5\right )\right )}{63 (d+e x)^4 \left (c d^2-a e^2\right )^6 \sqrt {(d+e x) (a e+c d x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
(-2*(7*a^5*e^10 - 5*a^4*c*d*e^8*(9*d + 2*e*x) + 2*a^3*c^2*d^2*e^6*(63*d^2 + 36*d*e*x + 8*e^2*x^2) - 2*a^2*c^3*
d^3*e^4*(105*d^3 + 126*d^2*e*x + 72*d*e^2*x^2 + 16*e^3*x^3) + a*c^4*d^4*e^2*(315*d^4 + 840*d^3*e*x + 1008*d^2*
e^2*x^2 + 576*d*e^3*x^3 + 128*e^4*x^4) + c^5*d^5*(63*d^5 + 630*d^4*e*x + 1680*d^3*e^2*x^2 + 2016*d^2*e^3*x^3 +
1152*d*e^4*x^4 + 256*e^5*x^5)))/(63*(c*d^2 - a*e^2)^6*(d + e*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)])
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IntegrateAlgebraic [F] time = 180.27, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[1/((d + e*x)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]
[Out]
$Aborted
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fricas [B] time = 40.31, size = 1004, normalized size = 3.34 \begin {gather*} -\frac {2 \, {\left (256 \, c^{5} d^{5} e^{5} x^{5} + 63 \, c^{5} d^{10} + 315 \, a c^{4} d^{8} e^{2} - 210 \, a^{2} c^{3} d^{6} e^{4} + 126 \, a^{3} c^{2} d^{4} e^{6} - 45 \, a^{4} c d^{2} e^{8} + 7 \, a^{5} e^{10} + 128 \, {\left (9 \, c^{5} d^{6} e^{4} + a c^{4} d^{4} e^{6}\right )} x^{4} + 32 \, {\left (63 \, c^{5} d^{7} e^{3} + 18 \, a c^{4} d^{5} e^{5} - a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 16 \, {\left (105 \, c^{5} d^{8} e^{2} + 63 \, a c^{4} d^{6} e^{4} - 9 \, a^{2} c^{3} d^{4} e^{6} + a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 2 \, {\left (315 \, c^{5} d^{9} e + 420 \, a c^{4} d^{7} e^{3} - 126 \, a^{2} c^{3} d^{5} e^{5} + 36 \, a^{3} c^{2} d^{3} e^{7} - 5 \, a^{4} c d e^{9}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{63 \, {\left (a c^{6} d^{17} e - 6 \, a^{2} c^{5} d^{15} e^{3} + 15 \, a^{3} c^{4} d^{13} e^{5} - 20 \, a^{4} c^{3} d^{11} e^{7} + 15 \, a^{5} c^{2} d^{9} e^{9} - 6 \, a^{6} c d^{7} e^{11} + a^{7} d^{5} e^{13} + {\left (c^{7} d^{13} e^{5} - 6 \, a c^{6} d^{11} e^{7} + 15 \, a^{2} c^{5} d^{9} e^{9} - 20 \, a^{3} c^{4} d^{7} e^{11} + 15 \, a^{4} c^{3} d^{5} e^{13} - 6 \, a^{5} c^{2} d^{3} e^{15} + a^{6} c d e^{17}\right )} x^{6} + {\left (5 \, c^{7} d^{14} e^{4} - 29 \, a c^{6} d^{12} e^{6} + 69 \, a^{2} c^{5} d^{10} e^{8} - 85 \, a^{3} c^{4} d^{8} e^{10} + 55 \, a^{4} c^{3} d^{6} e^{12} - 15 \, a^{5} c^{2} d^{4} e^{14} - a^{6} c d^{2} e^{16} + a^{7} e^{18}\right )} x^{5} + 5 \, {\left (2 \, c^{7} d^{15} e^{3} - 11 \, a c^{6} d^{13} e^{5} + 24 \, a^{2} c^{5} d^{11} e^{7} - 25 \, a^{3} c^{4} d^{9} e^{9} + 10 \, a^{4} c^{3} d^{7} e^{11} + 3 \, a^{5} c^{2} d^{5} e^{13} - 4 \, a^{6} c d^{3} e^{15} + a^{7} d e^{17}\right )} x^{4} + 10 \, {\left (c^{7} d^{16} e^{2} - 5 \, a c^{6} d^{14} e^{4} + 9 \, a^{2} c^{5} d^{12} e^{6} - 5 \, a^{3} c^{4} d^{10} e^{8} - 5 \, a^{4} c^{3} d^{8} e^{10} + 9 \, a^{5} c^{2} d^{6} e^{12} - 5 \, a^{6} c d^{4} e^{14} + a^{7} d^{2} e^{16}\right )} x^{3} + 5 \, {\left (c^{7} d^{17} e - 4 \, a c^{6} d^{15} e^{3} + 3 \, a^{2} c^{5} d^{13} e^{5} + 10 \, a^{3} c^{4} d^{11} e^{7} - 25 \, a^{4} c^{3} d^{9} e^{9} + 24 \, a^{5} c^{2} d^{7} e^{11} - 11 \, a^{6} c d^{5} e^{13} + 2 \, a^{7} d^{3} e^{15}\right )} x^{2} + {\left (c^{7} d^{18} - a c^{6} d^{16} e^{2} - 15 \, a^{2} c^{5} d^{14} e^{4} + 55 \, a^{3} c^{4} d^{12} e^{6} - 85 \, a^{4} c^{3} d^{10} e^{8} + 69 \, a^{5} c^{2} d^{8} e^{10} - 29 \, a^{6} c d^{6} e^{12} + 5 \, a^{7} d^{4} e^{14}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="fricas")
[Out]
-2/63*(256*c^5*d^5*e^5*x^5 + 63*c^5*d^10 + 315*a*c^4*d^8*e^2 - 210*a^2*c^3*d^6*e^4 + 126*a^3*c^2*d^4*e^6 - 45*
a^4*c*d^2*e^8 + 7*a^5*e^10 + 128*(9*c^5*d^6*e^4 + a*c^4*d^4*e^6)*x^4 + 32*(63*c^5*d^7*e^3 + 18*a*c^4*d^5*e^5 -
a^2*c^3*d^3*e^7)*x^3 + 16*(105*c^5*d^8*e^2 + 63*a*c^4*d^6*e^4 - 9*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + 2*
(315*c^5*d^9*e + 420*a*c^4*d^7*e^3 - 126*a^2*c^3*d^5*e^5 + 36*a^3*c^2*d^3*e^7 - 5*a^4*c*d*e^9)*x)*sqrt(c*d*e*x
^2 + a*d*e + (c*d^2 + a*e^2)*x)/(a*c^6*d^17*e - 6*a^2*c^5*d^15*e^3 + 15*a^3*c^4*d^13*e^5 - 20*a^4*c^3*d^11*e^7
+ 15*a^5*c^2*d^9*e^9 - 6*a^6*c*d^7*e^11 + a^7*d^5*e^13 + (c^7*d^13*e^5 - 6*a*c^6*d^11*e^7 + 15*a^2*c^5*d^9*e^
9 - 20*a^3*c^4*d^7*e^11 + 15*a^4*c^3*d^5*e^13 - 6*a^5*c^2*d^3*e^15 + a^6*c*d*e^17)*x^6 + (5*c^7*d^14*e^4 - 29*
a*c^6*d^12*e^6 + 69*a^2*c^5*d^10*e^8 - 85*a^3*c^4*d^8*e^10 + 55*a^4*c^3*d^6*e^12 - 15*a^5*c^2*d^4*e^14 - a^6*c
*d^2*e^16 + a^7*e^18)*x^5 + 5*(2*c^7*d^15*e^3 - 11*a*c^6*d^13*e^5 + 24*a^2*c^5*d^11*e^7 - 25*a^3*c^4*d^9*e^9 +
10*a^4*c^3*d^7*e^11 + 3*a^5*c^2*d^5*e^13 - 4*a^6*c*d^3*e^15 + a^7*d*e^17)*x^4 + 10*(c^7*d^16*e^2 - 5*a*c^6*d^
14*e^4 + 9*a^2*c^5*d^12*e^6 - 5*a^3*c^4*d^10*e^8 - 5*a^4*c^3*d^8*e^10 + 9*a^5*c^2*d^6*e^12 - 5*a^6*c*d^4*e^14
+ a^7*d^2*e^16)*x^3 + 5*(c^7*d^17*e - 4*a*c^6*d^15*e^3 + 3*a^2*c^5*d^13*e^5 + 10*a^3*c^4*d^11*e^7 - 25*a^4*c^3
*d^9*e^9 + 24*a^5*c^2*d^7*e^11 - 11*a^6*c*d^5*e^13 + 2*a^7*d^3*e^15)*x^2 + (c^7*d^18 - a*c^6*d^16*e^2 - 15*a^2
*c^5*d^14*e^4 + 55*a^3*c^4*d^12*e^6 - 85*a^4*c^3*d^10*e^8 + 69*a^5*c^2*d^8*e^10 - 29*a^6*c*d^6*e^12 + 5*a^7*d^
4*e^14)*x)
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 0.46Unable to transpose Error: Bad Argument Value
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maple [A] time = 0.06, size = 412, normalized size = 1.37 \begin {gather*} -\frac {2 \left (c d x +a e \right ) \left (256 c^{5} d^{5} e^{5} x^{5}+128 a \,c^{4} d^{4} e^{6} x^{4}+1152 c^{5} d^{6} e^{4} x^{4}-32 a^{2} c^{3} d^{3} e^{7} x^{3}+576 a \,c^{4} d^{5} e^{5} x^{3}+2016 c^{5} d^{7} e^{3} x^{3}+16 a^{3} c^{2} d^{2} e^{8} x^{2}-144 a^{2} c^{3} d^{4} e^{6} x^{2}+1008 a \,c^{4} d^{6} e^{4} x^{2}+1680 c^{5} d^{8} e^{2} x^{2}-10 a^{4} c d \,e^{9} x +72 a^{3} c^{2} d^{3} e^{7} x -252 a^{2} c^{3} d^{5} e^{5} x +840 a \,c^{4} d^{7} e^{3} x +630 c^{5} d^{9} e x +7 a^{5} e^{10}-45 a^{4} c \,d^{2} e^{8}+126 a^{3} c^{2} d^{4} e^{6}-210 a^{2} c^{3} d^{6} e^{4}+315 a \,c^{4} d^{8} e^{2}+63 c^{5} d^{10}\right )}{63 \left (e x +d \right )^{3} \left (a^{6} e^{12}-6 a^{5} c \,d^{2} e^{10}+15 a^{4} c^{2} d^{4} e^{8}-20 a^{3} c^{3} d^{6} e^{6}+15 a^{2} c^{4} d^{8} e^{4}-6 a \,c^{5} d^{10} e^{2}+c^{6} d^{12}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^4/(c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(3/2),x)
[Out]
-2/63*(c*d*x+a*e)*(256*c^5*d^5*e^5*x^5+128*a*c^4*d^4*e^6*x^4+1152*c^5*d^6*e^4*x^4-32*a^2*c^3*d^3*e^7*x^3+576*a
*c^4*d^5*e^5*x^3+2016*c^5*d^7*e^3*x^3+16*a^3*c^2*d^2*e^8*x^2-144*a^2*c^3*d^4*e^6*x^2+1008*a*c^4*d^6*e^4*x^2+16
80*c^5*d^8*e^2*x^2-10*a^4*c*d*e^9*x+72*a^3*c^2*d^3*e^7*x-252*a^2*c^3*d^5*e^5*x+840*a*c^4*d^7*e^3*x+630*c^5*d^9
*e*x+7*a^5*e^10-45*a^4*c*d^2*e^8+126*a^3*c^2*d^4*e^6-210*a^2*c^3*d^6*e^4+315*a*c^4*d^8*e^2+63*c^5*d^10)/(e*x+d
)^3/(a^6*e^12-6*a^5*c*d^2*e^10+15*a^4*c^2*d^4*e^8-20*a^3*c^3*d^6*e^6+15*a^2*c^4*d^8*e^4-6*a*c^5*d^10*e^2+c^6*d
^12)/(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(3/2)
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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(1/(e*x+d)^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),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 [B] time = 4.36, size = 3925, normalized size = 13.04
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((d + e*x)^4*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)),x)
[Out]
(((64*c^5*d^6*e)/(315*(a*e^2 - c*d^2)^7) + (16*c^4*d^4*e*(11*a*e^2 - 15*c*d^2))/(315*(a*e^2 - c*d^2)^7))*(x*(a
*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((e^2*(18*c^2*d^3 - 34*a*c*d*e^2))/(9*(a*e^2 - c*d^2)^2
*(7*a^2*e^5 + 7*c^2*d^4*e - 14*a*c*d^2*e^3)) + (16*c^2*d^3*e^2)/(9*(a*e^2 - c*d^2)^2*(7*a^2*e^5 + 7*c^2*d^4*e
- 14*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 - (((d*((64*c^5*d^6*e^3)/(315*(
a*e^2 - c*d^2)^6*(3*a*e^3 - 3*c*d^2*e)) + (8*c^4*d^4*e^3*(35*a*e^2 - 51*c*d^2))/(315*(a*e^2 - c*d^2)^6*(3*a*e^
3 - 3*c*d^2*e))))/e + (4*c^3*d^3*e^2*(47*c^2*d^4 - 39*a^2*e^4 + 8*a*c*d^2*e^2))/(315*(a*e^2 - c*d^2)^6*(3*a*e^
3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((d*((32*c^4*d^5*e^3)/(63*(a*e^
2 - c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)) + (4*c^3*d^3*e^3*(39*a*e^2 - 55*c*d^2))/(63*(a*e^2 -
c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3))))/e - (e^2*(126*c^4*d^6 - 472*a*c^3*d^4*e^2 + 314*a^2*c^2
*d^2*e^4))/(63*(a*e^2 - c*d^2)^4*(5*a^2*e^5 + 5*c^2*d^4*e - 10*a*c*d^2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d
*e*x^2)^(1/2))/(d + e*x)^3 + (((32*c^3*d^4*e^2)/(63*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)) - (16*c^2*d^2*e^2
*(a*e^2 + c*d^2))/(63*(a*e^2 - c*d^2)^4*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))
/(d + e*x)^3 + (((d*((d*((128*c^7*d^8*e^3)/(945*(a*e^2 - c*d^2)^9) + (16*c^6*d^6*e^3*(23*a*e^2 - 47*c*d^2))/(9
45*(a*e^2 - c*d^2)^9)))/e + (16*c^5*d^5*e^2*(109*a^2*e^4 + 179*c^2*d^4 - 264*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2
)^9)))/e - (4*c^4*d^4*e*(245*a^3*e^6 + 315*c^3*d^6 - 229*a*c^2*d^4*e^2 - 299*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d
^2)^9))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) - (((d*((d*((64*c^6*d^7*e^4)/(315*(a*e^2 - c*
d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3)) + (8*c^5*d^5*e^4*(9*a*e^2 - 17*c*d^2))/(105*(a*e^2 - c*d^2)^
6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (4*c^4*d^4*e^3*(245*a^2*e^4 + 401*c^2*d^4 - 598*a*c*d^2*e^2
))/(315*(a*e^2 - c*d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^2*e^3))))/e + (e^2*(630*c^6*d^9 - 3494*a*c^5*d^7*
e^2 + 4690*a^2*c^4*d^5*e^4 - 1890*a^3*c^3*d^3*e^6))/(315*(a*e^2 - c*d^2)^6*(3*a^2*e^5 + 3*c^2*d^4*e - 6*a*c*d^
2*e^3)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 - (2*e^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*
e*x^2)^(1/2))/((d + e*x)^5*(9*a^2*e^5 + 9*c^2*d^4*e - 18*a*c*d^2*e^3)) - ((x*((a*(((a*e^2 + c*d^2)*((128*c^9*d
^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*
(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*c^8
*d^8*e^4*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a
^2*c*d*e^5)) - (256*a*c^9*d^10*e^6)/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*
c^8*d^8*e^4*(a*e^2 + c*d^2)*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c
*d*e^5))))/c + ((a*e^2 + c*d^2)*((a*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a
*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*
c^2*d^3*e^3 + a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(
a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a
*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*c^8*d^8*e^4*(98*a^2*e^4 + 191*c^2
*d^4 - 241*a*c*d^2*e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (256*a*c^9*d^10
*e^6)/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^8*d^8*e^4*(a*e^2 + c*d^2)*(1
5*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*c^7*d
^7*e^3*(213*a^3*e^6 - 759*c^3*d^6 + 1513*a*c^2*d^4*e^2 - 1031*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*
e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (16*c^7*d^7*e^3*(a*e^2 + c*d^2)*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c*d^2*
e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (2*c^2*d^2*e^2*(1890*c^8
*d^12 - 13632*a*c^7*d^10*e^2 + 26500*a^2*c^6*d^8*e^4 - 20416*a^3*c^5*d^6*e^6 + 5530*a^4*c^4*d^4*e^8))/(945*(a*
e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (8*c^6*d^6*e^2*(a*e^2 + c*d^2)*(213*a^3*e^6 - 75
9*c^3*d^6 + 1513*a*c^2*d^4*e^2 - 1031*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^
2*c*d*e^5))) + (a*((a*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 +
a^2*c*d*e^5))))/c - ((a*e^2 + c*d^2)*(((a*e^2 + c*d^2)*((128*c^9*d^9*e^5*(a*e^2 + c*d^2))/(945*(a*e^2 - c*d^2)
^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) + (32*c^9*d^9*e^5*(15*a*e^2 - 47*c*d^2))/(945*(a*e^2 - c*d^2)^
8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) + (32*c^8*d^8*e^4*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c
*d^2*e^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (256*a*c^9*d^10*e^6)/(945*(a*
e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)) - (16*c^8*d^8*e^4*(a*e^2 + c*d^2)*(15*a*e^2 - 47*c
*d^2))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/(c*d*e) - (16*c^7*d^7*e^3*(213*a^
3*e^6 - 759*c^3*d^6 + 1513*a*c^2*d^4*e^2 - 1031*a^2*c*d^2*e^4))/(945*(a*e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^
3*e^3 + a^2*c*d*e^5)) + (16*c^7*d^7*e^3*(a*e^2 + c*d^2)*(98*a^2*e^4 + 191*c^2*d^4 - 241*a*c*d^2*e^2))/(945*(a*
e^2 - c*d^2)^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5))))/c + (c*d*e*(a*e^2 + c*d^2)*(1890*c^8*d^12 - 1363
2*a*c^7*d^10*e^2 + 26500*a^2*c^6*d^8*e^4 - 20416*a^3*c^5*d^6*e^6 + 5530*a^4*c^4*d^4*e^8))/(945*(a*e^2 - c*d^2)
^8*(c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/((a*e + c*d*x)
*(d + e*x)) + (64*c^4*d^4*e*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(315*(a*e^2 - c*d^2)^6*(d + e*x)) -
(32*c^3*d^3*e^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(105*(a*e^2 - c*d^2)^4*(3*a*e^3 - 3*c*d^2*e)*(
d + e*x)^2)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
Integral(1/(((d + e*x)*(a*e + c*d*x))**(3/2)*(d + e*x)**4), x)
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