3.2049 \(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=109 \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}} \]

[Out]

4/63*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/c^2/d^2/(e*x+d)^(7/2)+2/9*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(7/2)/c/d/(e*x+d)^(5/2)

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Rubi [A]  time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(4*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(63*c^2*d^2*(d + e*x)^(7/2)) + (2*(a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(9*c*d*(d + e*x)^(5/2))

Rule 648

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] /; 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 + p, 0]

Rule 656

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[(Simplify[m + p]*(2*c*d - b*e))/(c*(m + 2*p + 1)), In
t[(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] && E
qQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] && IGtQ[Simplify[m + p], 0]

Rubi steps

\begin {align*} \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{(d+e x)^{5/2}} \, dx}{9 d}\\ &=\frac {4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{63 c^2 d^2 (d+e x)^{7/2}}+\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{9 c d (d+e x)^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 65, normalized size = 0.60 \[ \frac {2 (a e+c d x)^3 \sqrt {(d+e x) (a e+c d x)} \left (c d (9 d+7 e x)-2 a e^2\right )}{63 c^2 d^2 \sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(-2*a*e^2 + c*d*(9*d + 7*e*x)))/(63*c^2*d^2*Sqrt[d + e*x])

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fricas [A]  time = 0.90, size = 169, normalized size = 1.55 \[ \frac {2 \, {\left (7 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} + {\left (9 \, c^{4} d^{5} + 19 \, a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{4} e + 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{63 \, {\left (c^{2} d^{2} e x + c^{2} d^{3}\right )}} \]

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)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

2/63*(7*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 2*a^4*e^5 + (9*c^4*d^5 + 19*a*c^3*d^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e +
5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqr
t(e*x + d)/(c^2*d^2*e*x + c^2*d^3)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \]

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)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(5/2)/(e*x + d)^(3/2), x)

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maple [A]  time = 0.05, size = 69, normalized size = 0.63 \[ -\frac {2 \left (c d x +a e \right ) \left (-7 c d e x +2 a \,e^{2}-9 c \,d^{2}\right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {5}{2}}}{63 \left (e x +d \right )^{\frac {5}{2}} c^{2} d^{2}} \]

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)^(5/2)/(e*x+d)^(3/2),x)

[Out]

-2/63*(c*d*x+a*e)*(-7*c*d*e*x+2*a*e^2-9*c*d^2)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/c^2/d^2/(e*x+d)^(5/2)

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maxima [A]  time = 1.31, size = 132, normalized size = 1.21 \[ \frac {2 \, {\left (7 \, c^{4} d^{4} e x^{4} + 9 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} + {\left (9 \, c^{4} d^{5} + 19 \, a c^{3} d^{3} e^{2}\right )} x^{3} + 3 \, {\left (9 \, a c^{3} d^{4} e + 5 \, a^{2} c^{2} d^{2} e^{3}\right )} x^{2} + {\left (27 \, a^{2} c^{2} d^{3} e^{2} + a^{3} c d e^{4}\right )} x\right )} \sqrt {c d x + a e}}{63 \, c^{2} d^{2}} \]

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)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

2/63*(7*c^4*d^4*e*x^4 + 9*a^3*c*d^2*e^3 - 2*a^4*e^5 + (9*c^4*d^5 + 19*a*c^3*d^3*e^2)*x^3 + 3*(9*a*c^3*d^4*e +
5*a^2*c^2*d^2*e^3)*x^2 + (27*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x)*sqrt(c*d*x + a*e)/(c^2*d^2)

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mupad [B]  time = 1.03, size = 156, normalized size = 1.43 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {x^3\,\left (18\,c^4\,d^5+38\,a\,c^3\,d^3\,e^2\right )}{63\,c^2\,d^2}-\frac {4\,a^4\,e^5-18\,a^3\,c\,d^2\,e^3}{63\,c^2\,d^2}+\frac {2\,c^2\,d^2\,e\,x^4}{9}+\frac {2\,a\,e\,x^2\,\left (9\,c\,d^2+5\,a\,e^2\right )}{21}+\frac {2\,a^2\,e^2\,x\,\left (27\,c\,d^2+a\,e^2\right )}{63\,c\,d}\right )}{\sqrt {d+e\,x}} \]

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)^(5/2)/(d + e*x)^(3/2),x)

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((x^3*(18*c^4*d^5 + 38*a*c^3*d^3*e^2))/(63*c^2*d^2) - (4*a^4*e^
5 - 18*a^3*c*d^2*e^3)/(63*c^2*d^2) + (2*c^2*d^2*e*x^4)/9 + (2*a*e*x^2*(5*a*e^2 + 9*c*d^2))/21 + (2*a^2*e^2*x*(
a*e^2 + 27*c*d^2))/(63*c*d)))/(d + e*x)^(1/2)

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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