### 3.600 $$\int \sqrt{d+e x} (a+c x^2)^2 \, dx$$

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

[Out]

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

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Rubi [A]  time = 0.0460433, 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} $\frac{4 c (d+e x)^{7/2} \left (a e^2+3 c d^2\right )}{7 e^5}-\frac{8 c d (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^5}+\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )^2}{3 e^5}+\frac{2 c^2 (d+e x)^{11/2}}{11 e^5}-\frac{8 c^2 d (d+e x)^{9/2}}{9 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]*(a + c*x^2)^2,x]

[Out]

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

Mathematica [A]  time = 0.0612854, size = 96, normalized size = 0.76 $\frac{2 (d+e x)^{3/2} \left (1155 a^2 e^4+66 a c e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+c^2 \left (240 d^2 e^2 x^2-192 d^3 e x+128 d^4-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*a^2*e^4 + 66*a*c*e^2*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + c^2*(128*d^4 - 192*d^3*e*x + 2
40*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4)))/(3465*e^5)

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Maple [A]  time = 0.046, size = 106, normalized size = 0.8 \begin{align*}{\frac{630\,{c}^{2}{x}^{4}{e}^{4}-560\,{c}^{2}d{x}^{3}{e}^{3}+1980\,ac{e}^{4}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-1584\,acd{e}^{3}x-384\,{c}^{2}{d}^{3}ex+2310\,{a}^{2}{e}^{4}+1056\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{3465\,{e}^{5}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(e*x+d)^(3/2)*(315*c^2*e^4*x^4-280*c^2*d*e^3*x^3+990*a*c*e^4*x^2+240*c^2*d^2*e^2*x^2-792*a*c*d*e^3*x-19
2*c^2*d^3*e*x+1155*a^2*e^4+528*a*c*d^2*e^2+128*c^2*d^4)/e^5

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Maxima [A]  time = 1.1602, size = 153, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} c^{2} - 1540 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{2} d + 990 \,{\left (3 \, c^{2} d^{2} + a c e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 2772 \,{\left (c^{2} d^{3} + a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 1155 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{3465 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*(e*x + d)^(11/2)*c^2 - 1540*(e*x + d)^(9/2)*c^2*d + 990*(3*c^2*d^2 + a*c*e^2)*(e*x + d)^(7/2) - 27
72*(c^2*d^3 + a*c*d*e^2)*(e*x + d)^(5/2) + 1155*(c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(e*x + d)^(3/2))/e^5

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Fricas [A]  time = 1.78311, size = 325, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (315 \, c^{2} e^{5} x^{5} + 35 \, c^{2} d e^{4} x^{4} + 128 \, c^{2} d^{5} + 528 \, a c d^{3} e^{2} + 1155 \, a^{2} d e^{4} - 10 \,{\left (4 \, c^{2} d^{2} e^{3} - 99 \, a c e^{5}\right )} x^{3} + 6 \,{\left (8 \, c^{2} d^{3} e^{2} + 33 \, a c d e^{4}\right )} x^{2} -{\left (64 \, c^{2} d^{4} e + 264 \, a c d^{2} e^{3} - 1155 \, a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(315*c^2*e^5*x^5 + 35*c^2*d*e^4*x^4 + 128*c^2*d^5 + 528*a*c*d^3*e^2 + 1155*a^2*d*e^4 - 10*(4*c^2*d^2*e^
3 - 99*a*c*e^5)*x^3 + 6*(8*c^2*d^3*e^2 + 33*a*c*d*e^4)*x^2 - (64*c^2*d^4*e + 264*a*c*d^2*e^3 - 1155*a^2*e^5)*x
)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 2.56615, size = 148, normalized size = 1.17 \begin{align*} \frac{2 \left (- \frac{4 c^{2} d \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{c^{2} \left (d + e x\right )^{\frac{11}{2}}}{11 e^{4}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (2 a c e^{2} + 6 c^{2} d^{2}\right )}{7 e^{4}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 4 a c d e^{2} - 4 c^{2} d^{3}\right )}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}\right )}{3 e^{4}}\right )}{e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-4*c**2*d*(d + e*x)**(9/2)/(9*e**4) + c**2*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(7/2)*(2*a*c*e**2 + 6*c
**2*d**2)/(7*e**4) + (d + e*x)**(5/2)*(-4*a*c*d*e**2 - 4*c**2*d**3)/(5*e**4) + (d + e*x)**(3/2)*(a**2*e**4 + 2
*a*c*d**2*e**2 + c**2*d**4)/(3*e**4))/e

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Giac [A]  time = 1.6526, size = 170, normalized size = 1.34 \begin{align*} \frac{2}{3465} \,{\left (66 \,{\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 e^{\left (-2\right )} +{\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} e^{\left (-4\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(66*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a*c*e^(-2) + (315*(x*e + d)^(1
1/2) - 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*e^(-4) + 1155*(x*e + d)^(3/2)*a^2)*e^(-1)