### 3.601 $$\int \frac{(a+c x^2)^2}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=125 $\frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5}$

[Out]

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

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Rubi [A]  time = 0.0453817, antiderivative size = 125, 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} $\frac{4 c (d+e x)^{5/2} \left (a e^2+3 c d^2\right )}{5 e^5}-\frac{8 c d (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^5}+\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )^2}{e^5}+\frac{2 c^2 (d+e x)^{9/2}}{9 e^5}-\frac{8 c^2 d (d+e x)^{7/2}}{7 e^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/Sqrt[d + e*x],x]

[Out]

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

Mathematica [A]  time = 0.0610542, size = 96, normalized size = 0.77 $\frac{2 \sqrt{d+e x} \left (315 a^2 e^4+42 a c e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^2 \left (48 d^2 e^2 x^2-64 d^3 e x+128 d^4-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(315*a^2*e^4 + 42*a*c*e^2*(8*d^2 - 4*d*e*x + 3*e^2*x^2) + c^2*(128*d^4 - 64*d^3*e*x + 48*d^2*
e^2*x^2 - 40*d*e^3*x^3 + 35*e^4*x^4)))/(315*e^5)

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Maple [A]  time = 0.045, size = 106, normalized size = 0.9 \begin{align*}{\frac{70\,{c}^{2}{x}^{4}{e}^{4}-80\,{c}^{2}d{x}^{3}{e}^{3}+252\,ac{e}^{4}{x}^{2}+96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}-336\,acd{e}^{3}x-128\,{c}^{2}{d}^{3}ex+630\,{a}^{2}{e}^{4}+672\,ac{d}^{2}{e}^{2}+256\,{c}^{2}{d}^{4}}{315\,{e}^{5}}\sqrt{ex+d}} \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]

2/315*(e*x+d)^(1/2)*(35*c^2*e^4*x^4-40*c^2*d*e^3*x^3+126*a*c*e^4*x^2+48*c^2*d^2*e^2*x^2-168*a*c*d*e^3*x-64*c^2
*d^3*e*x+315*a^2*e^4+336*a*c*d^2*e^2+128*c^2*d^4)/e^5

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Maxima [A]  time = 1.10894, size = 162, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} + \frac{42 \,{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} a c}{e^{2}} + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2}}{e^{4}}\right )}}{315 \, 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, algorithm="maxima")

[Out]

2/315*(315*sqrt(e*x + d)*a^2 + 42*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*c/e^2 +
(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x
+ d)*d^4)*c^2/e^4)/e

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Fricas [A]  time = 1.79864, size = 242, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (35 \, c^{2} e^{4} x^{4} - 40 \, c^{2} d e^{3} x^{3} + 128 \, c^{2} d^{4} + 336 \, a c d^{2} e^{2} + 315 \, a^{2} e^{4} + 6 \,{\left (8 \, c^{2} d^{2} e^{2} + 21 \, a c e^{4}\right )} x^{2} - 8 \,{\left (8 \, c^{2} d^{3} e + 21 \, a c d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, 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]

2/315*(35*c^2*e^4*x^4 - 40*c^2*d*e^3*x^3 + 128*c^2*d^4 + 336*a*c*d^2*e^2 + 315*a^2*e^4 + 6*(8*c^2*d^2*e^2 + 21
*a*c*e^4)*x^2 - 8*(8*c^2*d^3*e + 21*a*c*d*e^3)*x)*sqrt(e*x + d)/e^5

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Sympy [A]  time = 27.2914, size = 330, normalized size = 2.64 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{4 a c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 c^{2} d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{4}} + \frac{2 c^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + \frac{2 a c x^{3}}{3} + \frac{c^{2} x^{5}}{5}}{\sqrt{d}} & \text{otherwise} \end{cases} \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]

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

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Giac [A]  time = 1.23855, size = 170, normalized size = 1.36 \begin{align*} \frac{2}{315} \,{\left (42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a c e^{\left (-2\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} c^{2} e^{\left (-4\right )} + 315 \, \sqrt{x e + d} 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]

2/315*(42*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a*c*e^(-2) + (35*(x*e + d)^(9/2) -
180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*c^2*e^(-4)
+ 315*sqrt(x*e + d)*a^2)*e^(-1)