### 3.2046 $$\int (d+e x)^{3/2} (a d e+(c d^2+a e^2) x+c d e x^2)^{5/2} \, dx$$

Optimal. Leaf size=295 $\frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}$

[Out]

(256*(c*d^2 - a*e^2)^4*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(45045*c^5*d^5*(d + e*x)^(7/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*(d + e*x)^(5/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d^2 - a*e^2)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d)

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Rubi [A]  time = 0.265003, antiderivative size = 295, normalized size of antiderivative = 1., 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} $\frac{256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\frac{128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{15 c d}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(5/2),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)^(7/2))/(45045*c^5*d^5*(d + e*x)^(7/2)) + (128*(
c*d^2 - a*e^2)^3*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(6435*c^4*d^4*(d + e*x)^(5/2)) + (32*(c*d^2 -
a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(715*c^3*d^3*(d + e*x)^(3/2)) + (16*(c*d^2 - a*e^2)*(a
*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2))/(195*c^2*d^2*Sqrt[d + e*x]) + (2*Sqrt[d + e*x]*(a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2)^(7/2))/(15*c*d)

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]

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]

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx &=\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac{\left (8 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2} \, dx}{15 d}\\ &=\frac{16 \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}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac{\left (16 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx}{65 d^2}\\ &=\frac{32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \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}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac{\left (64 \left (d^2-\frac{a e^2}{c}\right )^3\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)^{3/2}} \, dx}{715 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 )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \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}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}+\frac{\left (128 \left (d^2-\frac{a e^2}{c}\right )^4\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}{6435 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 )^{7/2}}{45045 c^5 d^5 (d+e x)^{7/2}}+\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 )^{7/2}}{6435 c^4 d^4 (d+e x)^{5/2}}+\frac{32 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{715 c^3 d^3 (d+e x)^{3/2}}+\frac{16 \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}}{195 c^2 d^2 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{15 c d}\\ \end{align*}

Mathematica [A]  time = 0.153624, size = 197, normalized size = 0.67 $\frac{2 (a e+c d x)^3 \sqrt{(d+e x) (a e+c d x)} \left (48 a^2 c^2 d^2 e^4 \left (65 d^2+70 d e x+21 e^2 x^2\right )-64 a^3 c d e^6 (15 d+7 e x)+128 a^4 e^8-8 a c^3 d^3 e^2 \left (1365 d^2 e x+715 d^3+945 d e^2 x^2+231 e^3 x^3\right )+c^4 d^4 \left (24570 d^2 e^2 x^2+20020 d^3 e x+6435 d^4+13860 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 c^5 d^5 \sqrt{d+e x}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(a*e + c*d*x)^3*Sqrt[(a*e + c*d*x)*(d + e*x)]*(128*a^4*e^8 - 64*a^3*c*d*e^6*(15*d + 7*e*x) + 48*a^2*c^2*d^2
*e^4*(65*d^2 + 70*d*e*x + 21*e^2*x^2) - 8*a*c^3*d^3*e^2*(715*d^3 + 1365*d^2*e*x + 945*d*e^2*x^2 + 231*e^3*x^3)
+ c^4*d^4*(6435*d^4 + 20020*d^3*e*x + 24570*d^2*e^2*x^2 + 13860*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*c^5*d^5*Sq
rt[d + e*x])

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Maple [A]  time = 0.045, size = 243, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( 3003\,{e}^{4}{x}^{4}{c}^{4}{d}^{4}-1848\,a{c}^{3}{d}^{3}{e}^{5}{x}^{3}+13860\,{c}^{4}{d}^{5}{e}^{3}{x}^{3}+1008\,{a}^{2}{c}^{2}{d}^{2}{e}^{6}{x}^{2}-7560\,a{c}^{3}{d}^{4}{e}^{4}{x}^{2}+24570\,{c}^{4}{d}^{6}{e}^{2}{x}^{2}-448\,{a}^{3}cd{e}^{7}x+3360\,{a}^{2}{c}^{2}{d}^{3}{e}^{5}x-10920\,a{c}^{3}{d}^{5}{e}^{3}x+20020\,{c}^{4}{d}^{7}ex+128\,{a}^{4}{e}^{8}-960\,{a}^{3}c{d}^{2}{e}^{6}+3120\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-5720\,a{c}^{3}{d}^{6}{e}^{2}+6435\,{c}^{4}{d}^{8} \right ) }{45045\,{c}^{5}{d}^{5}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{5}{2}}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(c*d*x+a*e)*(3003*c^4*d^4*e^4*x^4-1848*a*c^3*d^3*e^5*x^3+13860*c^4*d^5*e^3*x^3+1008*a^2*c^2*d^2*e^6*x^
2-7560*a*c^3*d^4*e^4*x^2+24570*c^4*d^6*e^2*x^2-448*a^3*c*d*e^7*x+3360*a^2*c^2*d^3*e^5*x-10920*a*c^3*d^5*e^3*x+
20020*c^4*d^7*e*x+128*a^4*e^8-960*a^3*c*d^2*e^6+3120*a^2*c^2*d^4*e^4-5720*a*c^3*d^6*e^2+6435*c^4*d^8)*(c*d*e*x
^2+a*e^2*x+c*d^2*x+a*d*e)^(5/2)/c^5/d^5/(e*x+d)^(5/2)

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Maxima [A]  time = 1.20802, size = 605, normalized size = 2.05 \begin{align*} \frac{2 \,{\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \,{\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \,{\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \,{\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \,{\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \,{\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} +{\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\right )} x\right )} \sqrt{c d x + a e}{\left (e x + d\right )}}{45045 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c
*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7*d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^
4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636*a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)
*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)
*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 120*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d
^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 2860*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*
a^6*c*d*e^10)*x)*sqrt(c*d*x + a*e)*(e*x + d)/(c^5*d^5*e*x + c^5*d^6)

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Fricas [A]  time = 1.9239, size = 1026, normalized size = 3.48 \begin{align*} \frac{2 \,{\left (3003 \, c^{7} d^{7} e^{4} x^{7} + 6435 \, a^{3} c^{4} d^{8} e^{3} - 5720 \, a^{4} c^{3} d^{6} e^{5} + 3120 \, a^{5} c^{2} d^{4} e^{7} - 960 \, a^{6} c d^{2} e^{9} + 128 \, a^{7} e^{11} + 231 \,{\left (60 \, c^{7} d^{8} e^{3} + 31 \, a c^{6} d^{6} e^{5}\right )} x^{6} + 63 \,{\left (390 \, c^{7} d^{9} e^{2} + 540 \, a c^{6} d^{7} e^{4} + 71 \, a^{2} c^{5} d^{5} e^{6}\right )} x^{5} + 35 \,{\left (572 \, c^{7} d^{10} e + 1794 \, a c^{6} d^{8} e^{3} + 636 \, a^{2} c^{5} d^{6} e^{5} + a^{3} c^{4} d^{4} e^{7}\right )} x^{4} + 5 \,{\left (1287 \, c^{7} d^{11} + 10868 \, a c^{6} d^{9} e^{2} + 8814 \, a^{2} c^{5} d^{7} e^{4} + 60 \, a^{3} c^{4} d^{5} e^{6} - 8 \, a^{4} c^{3} d^{3} e^{8}\right )} x^{3} + 3 \,{\left (6435 \, a c^{6} d^{10} e + 14300 \, a^{2} c^{5} d^{8} e^{3} + 390 \, a^{3} c^{4} d^{6} e^{5} - 120 \, a^{4} c^{3} d^{4} e^{7} + 16 \, a^{5} c^{2} d^{2} e^{9}\right )} x^{2} +{\left (19305 \, a^{2} c^{5} d^{9} e^{2} + 2860 \, a^{3} c^{4} d^{7} e^{4} - 1560 \, a^{4} c^{3} d^{5} e^{6} + 480 \, a^{5} c^{2} d^{3} e^{8} - 64 \, a^{6} c d e^{10}\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}}{45045 \,{\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*c^7*d^7*e^4*x^7 + 6435*a^3*c^4*d^8*e^3 - 5720*a^4*c^3*d^6*e^5 + 3120*a^5*c^2*d^4*e^7 - 960*a^6*c
*d^2*e^9 + 128*a^7*e^11 + 231*(60*c^7*d^8*e^3 + 31*a*c^6*d^6*e^5)*x^6 + 63*(390*c^7*d^9*e^2 + 540*a*c^6*d^7*e^
4 + 71*a^2*c^5*d^5*e^6)*x^5 + 35*(572*c^7*d^10*e + 1794*a*c^6*d^8*e^3 + 636*a^2*c^5*d^6*e^5 + a^3*c^4*d^4*e^7)
*x^4 + 5*(1287*c^7*d^11 + 10868*a*c^6*d^9*e^2 + 8814*a^2*c^5*d^7*e^4 + 60*a^3*c^4*d^5*e^6 - 8*a^4*c^3*d^3*e^8)
*x^3 + 3*(6435*a*c^6*d^10*e + 14300*a^2*c^5*d^8*e^3 + 390*a^3*c^4*d^6*e^5 - 120*a^4*c^3*d^4*e^7 + 16*a^5*c^2*d
^2*e^9)*x^2 + (19305*a^2*c^5*d^9*e^2 + 2860*a^3*c^4*d^7*e^4 - 1560*a^4*c^3*d^5*e^6 + 480*a^5*c^2*d^3*e^8 - 64*
a^6*c*d*e^10)*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out