### 3.598 $$\int (d+e x)^{5/2} (a+c x^2)^2 \, dx$$

Optimal. Leaf size=127 $\frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5}$

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*e^5
)

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Rubi [A]  time = 0.0568408, 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} $\frac{4 c (d+e x)^{11/2} \left (a e^2+3 c d^2\right )}{11 e^5}-\frac{8 c d (d+e x)^{9/2} \left (a e^2+c d^2\right )}{9 e^5}+\frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )^2}{7 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2)^2,x]

[Out]

(2*(c*d^2 + a*e^2)^2*(d + e*x)^(7/2))/(7*e^5) - (8*c*d*(c*d^2 + a*e^2)*(d + e*x)^(9/2))/(9*e^5) + (4*c*(3*c*d^
2 + a*e^2)*(d + e*x)^(11/2))/(11*e^5) - (8*c^2*d*(d + e*x)^(13/2))/(13*e^5) + (2*c^2*(d + e*x)^(15/2))/(15*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)^{5/2} \left (a+c x^2\right )^2 \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2 (d+e x)^{5/2}}{e^4}-\frac{4 c d \left (c d^2+a e^2\right ) (d+e x)^{7/2}}{e^4}+\frac{2 c \left (3 c d^2+a e^2\right ) (d+e x)^{9/2}}{e^4}-\frac{4 c^2 d (d+e x)^{11/2}}{e^4}+\frac{c^2 (d+e x)^{13/2}}{e^4}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right )^2 (d+e x)^{7/2}}{7 e^5}-\frac{8 c d \left (c d^2+a e^2\right ) (d+e x)^{9/2}}{9 e^5}+\frac{4 c \left (3 c d^2+a e^2\right ) (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{13/2}}{13 e^5}+\frac{2 c^2 (d+e x)^{15/2}}{15 e^5}\\ \end{align*}

Mathematica [A]  time = 0.111509, size = 96, normalized size = 0.76 $\frac{2 (d+e x)^{7/2} \left (6435 a^2 e^4+130 a c e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+c^2 \left (1008 d^2 e^2 x^2-448 d^3 e x+128 d^4-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(6435*a^2*e^4 + 130*a*c*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + c^2*(128*d^4 - 448*d^3*e*x +
1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*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{6006\,{c}^{2}{x}^{4}{e}^{4}-3696\,{c}^{2}d{x}^{3}{e}^{3}+16380\,ac{e}^{4}{x}^{2}+2016\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-7280\,acd{e}^{3}x-896\,{c}^{2}{d}^{3}ex+12870\,{a}^{2}{e}^{4}+2080\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{45045\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(e*x+d)^(7/2)*(3003*c^2*e^4*x^4-1848*c^2*d*e^3*x^3+8190*a*c*e^4*x^2+1008*c^2*d^2*e^2*x^2-3640*a*c*d*e^
3*x-448*c^2*d^3*e*x+6435*a^2*e^4+1040*a*c*d^2*e^2+128*c^2*d^4)/e^5

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Maxima [A]  time = 1.12371, size = 153, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (3003 \,{\left (e x + d\right )}^{\frac{15}{2}} c^{2} - 13860 \,{\left (e x + d\right )}^{\frac{13}{2}} c^{2} d + 8190 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{11}{2}} - 20020 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 6435 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{45045 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45045*(3003*(e*x + d)^(15/2)*c^2 - 13860*(e*x + d)^(13/2)*c^2*d + 8190*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(11/2
) - 20020*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(9/2) + 6435*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(7/2))/e^
5

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Fricas [B]  time = 1.75913, size = 504, normalized size = 3.97 \begin{align*} \frac{2 \,{\left (3003 \, c^{2} e^{7} x^{7} + 7161 \, c^{2} d e^{6} x^{6} + 128 \, c^{2} d^{7} + 1040 \, a c d^{5} e^{2} + 6435 \, a^{2} d^{3} e^{4} + 63 \,{\left (71 \, c^{2} d^{2} e^{5} + 130 \, a c e^{7}\right )} x^{5} + 35 \,{\left (c^{2} d^{3} e^{4} + 598 \, a c d e^{6}\right )} x^{4} - 5 \,{\left (8 \, c^{2} d^{4} e^{3} - 2938 \, a c d^{2} e^{5} - 1287 \, a^{2} e^{7}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{5} e^{2} + 130 \, a c d^{3} e^{4} + 6435 \, a^{2} d e^{6}\right )} x^{2} -{\left (64 \, c^{2} d^{6} e + 520 \, a c d^{4} e^{3} - 19305 \, a^{2} d^{2} 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)^(5/2)*(c*x^2+a)^2,x, algorithm="fricas")

[Out]

2/45045*(3003*c^2*e^7*x^7 + 7161*c^2*d*e^6*x^6 + 128*c^2*d^7 + 1040*a*c*d^5*e^2 + 6435*a^2*d^3*e^4 + 63*(71*c^
2*d^2*e^5 + 130*a*c*e^7)*x^5 + 35*(c^2*d^3*e^4 + 598*a*c*d*e^6)*x^4 - 5*(8*c^2*d^4*e^3 - 2938*a*c*d^2*e^5 - 12
87*a^2*e^7)*x^3 + 3*(16*c^2*d^5*e^2 + 130*a*c*d^3*e^4 + 6435*a^2*d*e^6)*x^2 - (64*c^2*d^6*e + 520*a*c*d^4*e^3
- 19305*a^2*d^2*e^5)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 17.5292, size = 566, normalized size = 4.46 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**2*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
4*a*c*d**2*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 8*a*c*d*(-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 + 4*a*c*(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**3 + 2*c**2*d**2*(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 + 4*c**2*d*(-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 + 2*c**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**
3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**5

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Giac [B]  time = 1.71189, size = 676, normalized size = 5.32 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(5/2)*(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^2*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^2*e^(-4) + 15015*(x*e + d)^(3/2)*a^2*d^2 + 572*(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*d*e^(-2) + 10*(693*(x*e + d)^(13/2) - 4095*(x*e + d)^(1
1/2)*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*d*e^(-4) + 6006*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a^2*d + 26*(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)*a*c*e^(-2)
+ (3003*(x*e + d)^(15/2) - 20790*(x*e + d)^(13/2)*d + 61425*(x*e + d)^(11/2)*d^2 - 100100*(x*e + d)^(9/2)*d^3
+ 96525*(x*e + d)^(7/2)*d^4 - 54054*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6)*c^2*e^(-4) + 429*(15*(x*
e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a^2)*e^(-1)