### 3.593 $$\int \sqrt{d+e x} (a+c x^2) \, dx$$

Optimal. Leaf size=63 $\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}-\frac{4 c d (d+e x)^{5/2}}{5 e^3}$

[Out]

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

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Rubi [A]  time = 0.0207032, antiderivative size = 63, 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} $\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right )}{3 e^3}+\frac{2 c (d+e x)^{7/2}}{7 e^3}-\frac{4 c d (d+e x)^{5/2}}{5 e^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[d + e*x]*(a + c*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0310447, size = 44, normalized size = 0.7 $\frac{2 (d+e x)^{3/2} \left (35 a e^2+c \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(35*a*e^2 + c*(8*d^2 - 12*d*e*x + 15*e^2*x^2)))/(105*e^3)

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Maple [A]  time = 0.042, size = 41, normalized size = 0.7 \begin{align*}{\frac{30\,c{e}^{2}{x}^{2}-24\,cdex+70\,a{e}^{2}+16\,c{d}^{2}}{105\,{e}^{3}} \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)*(e*x+d)^(1/2),x)

[Out]

2/105*(e*x+d)^(3/2)*(15*c*e^2*x^2-12*c*d*e*x+35*a*e^2+8*c*d^2)/e^3

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Maxima [A]  time = 1.12747, size = 63, normalized size = 1. \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c - 42 \,{\left (e x + d\right )}^{\frac{5}{2}} c d + 35 \,{\left (c d^{2} + a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, 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="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*c - 42*(e*x + d)^(5/2)*c*d + 35*(c*d^2 + a*e^2)*(e*x + d)^(3/2))/e^3

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Fricas [A]  time = 1.66208, size = 143, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (15 \, c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 8 \, c d^{3} + 35 \, a d e^{2} -{\left (4 \, c d^{2} e - 35 \, a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, 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]

2/105*(15*c*e^3*x^3 + 3*c*d*e^2*x^2 + 8*c*d^3 + 35*a*d*e^2 - (4*c*d^2*e - 35*a*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 1.73479, size = 61, normalized size = 0.97 \begin{align*} \frac{2 \left (- \frac{2 c d \left (d + e x\right )^{\frac{5}{2}}}{5 e^{2}} + \frac{c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )}{3 e^{2}}\right )}{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)

[Out]

2*(-2*c*d*(d + e*x)**(5/2)/(5*e**2) + c*(d + e*x)**(7/2)/(7*e**2) + (d + e*x)**(3/2)*(a*e**2 + c*d**2)/(3*e**2
))/e

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Giac [A]  time = 1.32537, size = 74, normalized size = 1.17 \begin{align*} \frac{2}{105} \,{\left ({\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 )} c e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} 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]

2/105*((15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*c*e^(-2) + 35*(x*e + d)^(3/2)*a)*e
^(-1)