3.594 $$\int \frac{a+c x^2}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=61 $\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3}$

[Out]

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

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Rubi [A]  time = 0.0208208, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right )}{e^3}+\frac{2 c (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{3/2}}{3 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)/Sqrt[d + e*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0333469, size = 44, normalized size = 0.72 $\frac{2 \sqrt{d+e x} \left (15 a e^2+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(15*a*e^2 + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)))/(15*e^3)

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Maple [A]  time = 0.04, size = 41, normalized size = 0.7 \begin{align*}{\frac{6\,c{e}^{2}{x}^{2}-8\,cdex+30\,a{e}^{2}+16\,c{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(e*x+d)^(1/2)*(3*c*e^2*x^2-4*c*d*e*x+15*a*e^2+8*c*d^2)/e^3

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Maxima [A]  time = 1.14683, size = 72, normalized size = 1.18 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a + \frac{{\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 )} c}{e^{2}}\right )}}{15 \, e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(15*sqrt(e*x + d)*a + (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*c/e^2)/e

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Fricas [A]  time = 1.78838, size = 96, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (3 \, c e^{2} x^{2} - 4 \, c d e x + 8 \, c d^{2} + 15 \, a e^{2}\right )} \sqrt{e x + d}}{15 \, e^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*c*e^2*x^2 - 4*c*d*e*x + 8*c*d^2 + 15*a*e^2)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 5.98915, size = 150, normalized size = 2.46 \begin{align*} \begin{cases} - \frac{\frac{2 a d}{\sqrt{d + e x}} + 2 a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 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{2 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}}}{e} & \text{for}\: e \neq 0 \\\frac{a x + \frac{c x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-(2*a*d/sqrt(d + e*x) + 2*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) + 2*c*d*(d**2/sqrt(d + e*x) + 2*d*sq
rt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*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)/e, Ne(e, 0)), ((a*x + c*x**3/3)/sqrt(d), True))

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Giac [A]  time = 1.38449, size = 74, normalized size = 1.21 \begin{align*} \frac{2}{15} \,{\left ({\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 )} c e^{\left (-2\right )} + 15 \, \sqrt{x e + d} a\right )} e^{\left (-1\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*((3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*c*e^(-2) + 15*sqrt(x*e + d)*a)*e^(-1)