3.1916 \(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^5} \, dx\)

Optimal. Leaf size=171 \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )} \]

[Out]

2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)/(e*x+d)^5+8/35*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(3/2)/(-a*e^2+c*d^2)^2/(e*x+d)^4+16/105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^3/(e*
x+d)^3

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Rubi [A]  time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {658, 650} \[ \frac {16 c^2 d^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 (d+e x)^3 \left (c d^2-a e^2\right )^3}+\frac {8 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{35 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^5) + (8*c*d*(a*d*e + (c*d^2 + a
*e^2)*x + c*d*e*x^2)^(3/2))/(35*(c*d^2 - a*e^2)^2*(d + e*x)^4) + (16*c^2*d^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^3)

Rule 650

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))/((p + 1)*(2*c*d - b*e)), 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] && EqQ[m + 2*p + 2, 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 {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {(4 c d) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^4} \, dx}{7 \left (c d^2-a e^2\right )}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {\left (8 c^2 d^2\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^3} \, dx}{35 \left (c d^2-a e^2\right )^2}\\ &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 124, normalized size = 0.73 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)} \left (15 a^3 e^5+3 a^2 c d e^3 (e x-14 d)+a c^2 d^2 e \left (35 d^2-14 d e x-4 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )}{105 (d+e x)^4 \left (c d^2-a e^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(15*a^3*e^5 + 3*a^2*c*d*e^3*(-14*d + e*x) + a*c^2*d^2*e*(35*d^2 - 14*d*e*x -
4*e^2*x^2) + c^3*d^3*x*(35*d^2 + 28*d*e*x + 8*e^2*x^2)))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^4)

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fricas [B]  time = 6.58, size = 373, normalized size = 2.18 \[ \frac {2 \, {\left (8 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 42 \, a^{2} c d^{2} e^{3} + 15 \, a^{3} e^{5} + 4 \, {\left (7 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{5} - 14 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{105 \, {\left (c^{3} d^{10} - 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} - a^{3} d^{4} e^{6} + {\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="fricas")

[Out]

2/105*(8*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 42*a^2*c*d^2*e^3 + 15*a^3*e^5 + 4*(7*c^3*d^4*e - a*c^2*d^2*e^3)*x^
2 + (35*c^3*d^5 - 14*a*c^2*d^3*e^2 + 3*a^2*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^10 -
3*a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4 - a^3*d^4*e^6 + (c^3*d^6*e^4 - 3*a*c^2*d^4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10
)*x^4 + 4*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4
+ 3*a^2*c*d^4*e^6 - a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2*c*d^5*e^5 - a^3*d^3*e^7)*x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: exp(1)*(2*(-(exp(1)*x+d)^-1/exp(1)*(-(ex
p(1)*x+d)^-1/exp(1)*(-(-3*c^3*d^5*sign((exp(1)*x+d)^-1)*exp(1)^3+6*a*c^2*d^3*sign((exp(1)*x+d)^-1)*exp(1)^5-3*
a^2*c*d*sign((exp(1)*x+d)^-1)*exp(1)^7)/(-105*a^3*exp(1)^10+105*c^3*d^6*exp(1)^4-315*a*c^2*d^4*exp(1)^6+315*a^
2*c*d^2*exp(1)^8)+(exp(1)*x+d)^-1/exp(1)*(15*a^3*sign((exp(1)*x+d)^-1)*exp(1)^10-15*c^3*d^6*sign((exp(1)*x+d)^
-1)*exp(1)^4+45*a*c^2*d^4*sign((exp(1)*x+d)^-1)*exp(1)^6-45*a^2*c*d^2*sign((exp(1)*x+d)^-1)*exp(1)^8)/(-105*a^
3*exp(1)^10+105*c^3*d^6*exp(1)^4-315*a*c^2*d^4*exp(1)^6+315*a^2*c*d^2*exp(1)^8))-(4*c^3*d^4*sign((exp(1)*x+d)^
-1)*exp(2)-4*a*c^2*d^2*sign((exp(1)*x+d)^-1)*exp(1)^4)/(-105*a^3*exp(1)^10+105*c^3*d^6*exp(1)^4-315*a*c^2*d^4*
exp(1)^6+315*a^2*c*d^2*exp(1)^8))-C_0*(210*a^2*exp(1)^6+210*c^2*d^4*exp(2)-420*a*c*d^2*exp(1)^4)/(-105*a^3*exp
(1)^10+105*c^3*d^6*exp(1)^4-315*a*c^2*d^4*exp(1)^6+315*a^2*c*d^2*exp(1)^8)+8*c^3*d^3*sign((exp(1)*x+d)^-1)*exp
(1)/(-105*a^3*exp(1)^10+105*c^3*d^6*exp(1)^4-315*a*c^2*d^4*exp(1)^6+315*a^2*c*d^2*exp(1)^8))*sqrt(c*d*exp(1)+a
*d*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^5+a*(exp(1)*x+d)^-1/exp(1)*exp(1)^2*exp(2)-c*d^2*(exp(1)*x+d)^-1/exp(1)*
exp(1)^2-a*d*(-(exp(1)*x+d)^-1/exp(1))^2*exp(1)^3*exp(2))-4*C_0*sqrt(a*d*exp(1)^3-a*d*exp(1)*exp(2))*ln(abs(a*
sqrt(a*d*exp(1)^3-a*d*exp(1)*exp(2))*exp(2)-c*d^2*sqrt(a*d*exp(1)^3-a*d*exp(1)*exp(2))))/(2*a*d*exp(1)^4-2*a*d
*exp(2)^2)+16*c^3*d^3*sqrt(c*d*exp(1))/(105*a^3*exp(1)^9-315*a^2*c*d^2*exp(1)^7+315*a*c^2*d^4*exp(1)^5-105*c^3
*d^6*exp(1)^3)*sign((exp(1)*x+d)^-1))

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maple [A]  time = 0.06, size = 146, normalized size = 0.85 \[ -\frac {2 \left (c d x +a e \right ) \left (8 c^{2} d^{2} e^{2} x^{2}-12 a c d \,e^{3} x +28 c^{2} d^{3} e x +15 a^{2} e^{4}-42 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (e x +d \right )^{4} \left (a^{3} e^{6}-3 a^{2} c \,d^{2} e^{4}+3 a \,c^{2} d^{4} e^{2}-c^{3} d^{6}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*e*x^2+a*d*e+(a*e^2+c*d^2)*x)^(1/2)/(e*x+d)^5,x)

[Out]

-2/105*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-12*a*c*d*e^3*x+28*c^2*d^3*e*x+15*a^2*e^4-42*a*c*d^2*e^2+35*c^2*d^4)*(c*d
*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(e*x+d)^4/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)

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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((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^5,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 = 1.95, size = 877, normalized size = 5.13 \[ \frac {\left (\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {2\,a\,e^2}{7\,a\,e^3-7\,c\,d^2\,e}-\frac {2\,c\,d^2}{7\,a\,e^3-7\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {8\,c^4\,d^5+8\,a\,c^3\,d^3\,e^2}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,a\,c^3\,d^3\,e}{105\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {12\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (d+e\,x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)/(d + e*x)^5,x)

[Out]

(((4*c^2*d^3)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*a*c*d*e^2)/(7*(a*e^2 - c*d^2)*(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 - (((2*a*e^2)/(7*a*e^3 - 7*c*d^2*e) - (2*c*d^2
)/(7*a*e^3 - 7*c*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 + (((4*c^3*d^4 + 4*a*c^2*d
^2*e^2)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^3*d^4)/(35*(a*e^2 - c*d^2)^2*(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 + (((8*c^4*d^5 + 8*a*c^3*d^3*e^2)/(105*e*(a*e^2
- c*d^2)^4) - (16*c^4*d^5)/(105*e*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)
+ (((2*c^2*d^3 + 2*a*c*d*e^2)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*c^2*d^3)/(7*(a*e^2 - c*d^2)*(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 + (((8*c^3*d^4)/(35*(a*e^2 - c*
d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*a*c^2*d^2*e^2)/(35*(a*e^2 - c*d^2)^2*(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 + (((16*c^4*d^5)/(105*e*(a*e^2 - c*d^2)^4) - (16*a*c^3*d^3*e)/(10
5*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (12*c^2*d^2*(x*(a*e^2 + c*d^2
) + a*d*e + c*d*e*x^2)^(1/2))/(35*(a*e^2 - c*d^2)*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) - (8*c^3*d^3*(x*(a*e^2 +
c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(105*e*(a*e^2 - c*d^2)^3*(d + e*x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**5,x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**5, x)

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