3.2027 \(\int (d+e x)^{7/2} \sqrt {a d e+(c d^2+a e^2) x+c d e x^2} \, dx\)

Optimal. Leaf size=295 \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]

[Out]

256/3465*(-a*e^2+c*d^2)^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^5/d^5/(e*x+d)^(3/2)+16/99*(-a*e^2+c*d^2)*(
e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^2/d^2+2/11*(e*x+d)^(5/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(3/2)/c/d+128/1155*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/(e*x+d)^(1/2)+32/231*(
-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*(e*x+d)^(1/2)/c^3/d^3

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Rubi [A]  time = 0.26, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(3465*c^5*d^5*(d + e*x)^(3/2)) + (128*(c
*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(1155*c^4*d^4*Sqrt[d + e*x]) + (32*(c*d^2 - a*e
^2)^2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(231*c^3*d^3) + (16*(c*d^2 - a*e^2)*(d + e*
x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(99*c^2*d^2) + (2*(d + e*x)^(5/2)*(a*d*e + (c*d^2 + a*
e^2)*x + c*d*e*x^2)^(3/2))/(11*c*d)

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 (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac {2 (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}}{11 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{11 d}\\ &=\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (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}}{11 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{33 d^2}\\ &=\frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (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}}{11 c d}+\frac {\left (64 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{231 d^3}\\ &=\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (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}}{11 c d}+\frac {\left (128 \left (d^2-\frac {a e^2}{c}\right )^4\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{1155 d^4}\\ &=\frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (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}}{11 c d}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 187, normalized size = 0.63 \[ \frac {2 ((d+e x) (a e+c d x))^{3/2} \left (128 a^4 e^8-64 a^3 c d e^6 (11 d+3 e x)+48 a^2 c^2 d^2 e^4 \left (33 d^2+22 d e x+5 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (231 d^3+297 d^2 e x+165 d e^2 x^2+35 e^3 x^3\right )+c^4 d^4 \left (1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(128*a^4*e^8 - 64*a^3*c*d*e^6*(11*d + 3*e*x) + 48*a^2*c^2*d^2*e^4*(33*d^2 +
22*d*e*x + 5*e^2*x^2) - 8*a*c^3*d^3*e^2*(231*d^3 + 297*d^2*e*x + 165*d*e^2*x^2 + 35*e^3*x^3) + c^4*d^4*(1155*
d^4 + 2772*d^3*e*x + 2970*d^2*e^2*x^2 + 1540*d*e^3*x^3 + 315*e^4*x^4)))/(3465*c^5*d^5*(d + e*x)^(3/2))

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fricas [A]  time = 0.92, size = 315, normalized size = 1.07 \[ \frac {2 \, {\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \, {\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c^5*d^5*e^4*x^5 + 1155*a*c^4*d^8*e - 1848*a^2*c^3*d^6*e^3 + 1584*a^3*c^2*d^4*e^5 - 704*a^4*c*d^2*e
^7 + 128*a^5*e^9 + 35*(44*c^5*d^6*e^3 + a*c^4*d^4*e^5)*x^4 + 10*(297*c^5*d^7*e^2 + 22*a*c^4*d^5*e^4 - 4*a^2*c^
3*d^3*e^6)*x^3 + 6*(462*c^5*d^8*e + 99*a*c^4*d^6*e^3 - 44*a^2*c^3*d^4*e^5 + 8*a^3*c^2*d^2*e^7)*x^2 + (1155*c^5
*d^9 + 924*a*c^4*d^7*e^2 - 792*a^2*c^3*d^5*e^4 + 352*a^3*c^2*d^3*e^6 - 64*a^4*c*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (e x + d\right )}^{\frac {7}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.04, size = 243, normalized size = 0.82 \[ \frac {2 \left (c d x +a e \right ) \left (315 c^{4} d^{4} e^{4} x^{4}-280 a \,c^{3} d^{3} e^{5} x^{3}+1540 c^{4} d^{5} e^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{6} x^{2}-1320 a \,c^{3} d^{4} e^{4} x^{2}+2970 c^{4} d^{6} e^{2} x^{2}-192 a^{3} c d \,e^{7} x +1056 a^{2} c^{2} d^{3} e^{5} x -2376 a \,c^{3} d^{5} e^{3} x +2772 c^{4} d^{7} e x +128 a^{4} e^{8}-704 a^{3} c \,d^{2} e^{6}+1584 a^{2} c^{2} d^{4} e^{4}-1848 a \,c^{3} d^{6} e^{2}+1155 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3465 \sqrt {e x +d}\, c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(c*d*x+a*e)*(315*c^4*d^4*e^4*x^4-280*a*c^3*d^3*e^5*x^3+1540*c^4*d^5*e^3*x^3+240*a^2*c^2*d^2*e^6*x^2-132
0*a*c^3*d^4*e^4*x^2+2970*c^4*d^6*e^2*x^2-192*a^3*c*d*e^7*x+1056*a^2*c^2*d^3*e^5*x-2376*a*c^3*d^5*e^3*x+2772*c^
4*d^7*e*x+128*a^4*e^8-704*a^3*c*d^2*e^6+1584*a^2*c^2*d^4*e^4-1848*a*c^3*d^6*e^2+1155*c^4*d^8)*(c*d*e*x^2+a*e^2
*x+c*d^2*x+a*d*e)^(1/2)/c^5/d^5/(e*x+d)^(1/2)

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maxima [A]  time = 1.54, size = 296, normalized size = 1.00 \[ \frac {2 \, {\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \, {\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c^5*d^5*e^4*x^5 + 1155*a*c^4*d^8*e - 1848*a^2*c^3*d^6*e^3 + 1584*a^3*c^2*d^4*e^5 - 704*a^4*c*d^2*e
^7 + 128*a^5*e^9 + 35*(44*c^5*d^6*e^3 + a*c^4*d^4*e^5)*x^4 + 10*(297*c^5*d^7*e^2 + 22*a*c^4*d^5*e^4 - 4*a^2*c^
3*d^3*e^6)*x^3 + 6*(462*c^5*d^8*e + 99*a*c^4*d^6*e^3 - 44*a^2*c^3*d^4*e^5 + 8*a^3*c^2*d^2*e^7)*x^2 + (1155*c^5
*d^9 + 924*a*c^4*d^7*e^2 - 792*a^2*c^3*d^5*e^4 + 352*a^3*c^2*d^3*e^6 - 64*a^4*c*d*e^8)*x)*sqrt(c*d*x + a*e)*(e
*x + d)/(c^5*d^5*e*x + c^5*d^6)

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mupad [B]  time = 1.21, size = 346, normalized size = 1.17 \[ \frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^3\,x^5\,\sqrt {d+e\,x}}{11}+\frac {4\,x^2\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-44\,a^2\,c\,d^2\,e^4+99\,a\,c^2\,d^4\,e^2+462\,c^3\,d^6\right )}{1155\,c^3\,d^3}+\frac {\sqrt {d+e\,x}\,\left (256\,a^5\,e^9-1408\,a^4\,c\,d^2\,e^7+3168\,a^3\,c^2\,d^4\,e^5-3696\,a^2\,c^3\,d^6\,e^3+2310\,a\,c^4\,d^8\,e\right )}{3465\,c^5\,d^5\,e}+\frac {2\,e^2\,x^4\,\left (44\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{99\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^4\,c\,d\,e^8+704\,a^3\,c^2\,d^3\,e^6-1584\,a^2\,c^3\,d^5\,e^4+1848\,a\,c^4\,d^7\,e^2+2310\,c^5\,d^9\right )}{3465\,c^5\,d^5\,e}+\frac {4\,e\,x^3\,\sqrt {d+e\,x}\,\left (-4\,a^2\,e^4+22\,a\,c\,d^2\,e^2+297\,c^2\,d^4\right )}{693\,c^2\,d^2}\right )}{x+\frac {d}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e^3*x^5*(d + e*x)^(1/2))/11 + (4*x^2*(d + e*x)^(1/2)*(8*a^3
*e^6 + 462*c^3*d^6 + 99*a*c^2*d^4*e^2 - 44*a^2*c*d^2*e^4))/(1155*c^3*d^3) + ((d + e*x)^(1/2)*(256*a^5*e^9 - 14
08*a^4*c*d^2*e^7 - 3696*a^2*c^3*d^6*e^3 + 3168*a^3*c^2*d^4*e^5 + 2310*a*c^4*d^8*e))/(3465*c^5*d^5*e) + (2*e^2*
x^4*(a*e^2 + 44*c*d^2)*(d + e*x)^(1/2))/(99*c*d) + (x*(d + e*x)^(1/2)*(2310*c^5*d^9 + 1848*a*c^4*d^7*e^2 - 158
4*a^2*c^3*d^5*e^4 + 704*a^3*c^2*d^3*e^6 - 128*a^4*c*d*e^8))/(3465*c^5*d^5*e) + (4*e*x^3*(d + e*x)^(1/2)*(297*c
^2*d^4 - 4*a^2*e^4 + 22*a*c*d^2*e^2))/(693*c^2*d^2)))/(x + d/e)

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

[Out]

Timed out

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