### 3.599 $$\int (d+e x)^{3/2} (a+c x^2)^2 \, dx$$

Optimal. Leaf size=127 $\frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5}$

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d + e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

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Rubi [A]  time = 0.0476634, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {697} $\frac{4 c (d+e x)^{9/2} \left (a e^2+3 c d^2\right )}{9 e^5}-\frac{8 c d (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^5}+\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )^2}{5 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(3/2)*(a + c*x^2)^2,x]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(5/2))/(5*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(7/2))/(7*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) - (8*c^2*d*(d + e*x)^(11/2))/(11*e^5) + (2*c^2*(d + e*x)^(13/2))/(13*e^5)

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}-\frac{4 c^2 d (d+e x)^{9/2}}{e^4}+\frac{c^2 (d+e x)^{11/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{5 e^5}-\frac{8 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{7 e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{11/2}}{11 e^5}+\frac{2 c^2 (d+e x)^{13/2}}{13 e^5}\\ \end{align*}

Mathematica [A]  time = 0.0870545, size = 97, normalized size = 0.76 $\frac{2 (d+e x)^{5/2} \left (9009 a^2 e^4+286 a c e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+3 c^2 \left (560 d^2 e^2 x^2-320 d^3 e x+128 d^4-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{45045 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(5/2)*(9009*a^2*e^4 + 286*a*c*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 3*c^2*(128*d^4 - 320*d^3*e*x
+ 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(45045*e^5)

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Maple [A]  time = 0.044, size = 106, normalized size = 0.8 \begin{align*}{\frac{6930\,{c}^{2}{x}^{4}{e}^{4}-5040\,{c}^{2}d{x}^{3}{e}^{3}+20020\,ac{e}^{4}{x}^{2}+3360\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-11440\,acd{e}^{3}x-1920\,{c}^{2}{d}^{3}ex+18018\,{a}^{2}{e}^{4}+4576\,ac{d}^{2}{e}^{2}+768\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \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)*(c*x^2+a)^2,x)

[Out]

2/45045*(e*x+d)^(5/2)*(3465*c^2*e^4*x^4-2520*c^2*d*e^3*x^3+10010*a*c*e^4*x^2+1680*c^2*d^2*e^2*x^2-5720*a*c*d*e
^3*x-960*c^2*d^3*e*x+9009*a^2*e^4+2288*a*c*d^2*e^2+384*c^2*d^4)/e^5

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Maxima [A]  time = 1.14843, size = 153, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (3465 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} - 16380 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} d + 10010 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} - 25740 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 9009 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*(e*x + d)^(13/2)*c^2 - 16380*(e*x + d)^(11/2)*c^2*d + 10010*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(9/2
) - 25740*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(7/2) + 9009*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(5/2))/e^
5

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Fricas [A]  time = 1.81298, size = 423, normalized size = 3.33 \begin{align*} \frac{2 \,{\left (3465 \, c^{2} e^{6} x^{6} + 4410 \, c^{2} d e^{5} x^{5} + 384 \, c^{2} d^{6} + 2288 \, a c d^{4} e^{2} + 9009 \, a^{2} d^{2} e^{4} + 35 \,{\left (3 \, c^{2} d^{2} e^{4} + 286 \, a c e^{6}\right )} x^{4} - 20 \,{\left (6 \, c^{2} d^{3} e^{3} - 715 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (48 \, c^{2} d^{4} e^{2} + 286 \, a c d^{2} e^{4} + 3003 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (96 \, c^{2} d^{5} e + 572 \, a c d^{3} e^{3} - 9009 \, a^{2} d e^{5}\right )} x\right )} \sqrt{e x + d}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3465*c^2*e^6*x^6 + 4410*c^2*d*e^5*x^5 + 384*c^2*d^6 + 2288*a*c*d^4*e^2 + 9009*a^2*d^2*e^4 + 35*(3*c^2
*d^2*e^4 + 286*a*c*e^6)*x^4 - 20*(6*c^2*d^3*e^3 - 715*a*c*d*e^5)*x^3 + 3*(48*c^2*d^4*e^2 + 286*a*c*d^2*e^4 + 3
003*a^2*e^6)*x^2 - 2*(96*c^2*d^5*e + 572*a*c*d^3*e^3 - 9009*a^2*d*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 10.6265, size = 328, normalized size = 2.58 \begin{align*} a^{2} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{4 a c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 c^{2} d \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{5}} + \frac{2 c^{2} \left (- \frac{d^{5} \left (d + e x\right )^{\frac{3}{2}}}{3} + d^{4} \left (d + e x\right )^{\frac{5}{2}} - \frac{10 d^{3} \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{10 d^{2} \left (d + e x\right )^{\frac{9}{2}}}{9} - \frac{5 d \left (d + e x\right )^{\frac{11}{2}}}{11} + \frac{\left (d + e x\right )^{\frac{13}{2}}}{13}\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*a**2*(-d*(d + e*x)**(3/2)/3 + (d
+ e*x)**(5/2)/5)/e + 4*a*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 4
*a*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3
+ 2*c**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**
(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 2*c**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*d**3*(d +
e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**5

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Giac [B]  time = 1.41704, size = 394, normalized size = 3.1 \begin{align*} \frac{2}{45045} \,{\left (858 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a c d e^{\left (-2\right )} + 13 \,{\left (315 \,{\left (x e + d\right )}^{\frac{11}{2}} - 1540 \,{\left (x e + d\right )}^{\frac{9}{2}} d + 2970 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{2} - 2772 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{3} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{4}\right )} c^{2} d e^{\left (-4\right )} + 15015 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d + 286 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} a c e^{\left (-2\right )} + 5 \,{\left (693 \,{\left (x e + d\right )}^{\frac{13}{2}} - 4095 \,{\left (x e + d\right )}^{\frac{11}{2}} d + 10010 \,{\left (x e + d\right )}^{\frac{9}{2}} d^{2} - 12870 \,{\left (x e + d\right )}^{\frac{7}{2}} d^{3} + 9009 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{4} - 3003 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{5}\right )} c^{2} e^{\left (-4\right )} + 3003 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(858*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c*d*e^(-2) + 13*(315*(x*e
+ d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 2970*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3
/2)*d^4)*c^2*d*e^(-4) + 15015*(x*e + d)^(3/2)*a^2*d + 286*(35*(x*e + d)^(9/2) - 135*(x*e + d)^(7/2)*d + 189*(x
*e + d)^(5/2)*d^2 - 105*(x*e + d)^(3/2)*d^3)*a*c*e^(-2) + 5*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(11/2)*d +
10010*(x*e + d)^(9/2)*d^2 - 12870*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 3003*(x*e + d)^(3/2)*d^5)*c
^2*e^(-4) + 3003*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2)*e^(-1)