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

Optimal. Leaf size=63 \[ \frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}-\frac{4 c d (d+e x)^{9/2}}{9 e^3} \]

[Out]

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

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Rubi [A]  time = 0.0238877, 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} \[ \frac{2 (d+e x)^{7/2} \left (a e^2+c d^2\right )}{7 e^3}+\frac{2 c (d+e x)^{11/2}}{11 e^3}-\frac{4 c d (d+e x)^{9/2}}{9 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(5/2)*(a + c*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0491549, size = 44, normalized size = 0.7 \[ \frac{2 (d+e x)^{7/2} \left (99 a e^2+c \left (8 d^2-28 d e x+63 e^2 x^2\right )\right )}{693 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(7/2)*(99*a*e^2 + c*(8*d^2 - 28*d*e*x + 63*e^2*x^2)))/(693*e^3)

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Maple [A]  time = 0.043, size = 41, normalized size = 0.7 \begin{align*}{\frac{126\,c{e}^{2}{x}^{2}-56\,cdex+198\,a{e}^{2}+16\,c{d}^{2}}{693\,{e}^{3}} \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),x)

[Out]

2/693*(e*x+d)^(7/2)*(63*c*e^2*x^2-28*c*d*e*x+99*a*e^2+8*c*d^2)/e^3

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Maxima [A]  time = 1.15476, size = 63, normalized size = 1. \begin{align*} \frac{2 \,{\left (63 \,{\left (e x + d\right )}^{\frac{11}{2}} c - 154 \,{\left (e x + d\right )}^{\frac{9}{2}} c d + 99 \,{\left (c d^{2} + a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*(e*x + d)^(11/2)*c - 154*(e*x + d)^(9/2)*c*d + 99*(c*d^2 + a*e^2)*(e*x + d)^(7/2))/e^3

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Fricas [B]  time = 1.91704, size = 244, normalized size = 3.87 \begin{align*} \frac{2 \,{\left (63 \, c e^{5} x^{5} + 161 \, c d e^{4} x^{4} + 8 \, c d^{5} + 99 \, a d^{3} e^{2} +{\left (113 \, c d^{2} e^{3} + 99 \, a e^{5}\right )} x^{3} + 3 \,{\left (c d^{3} e^{2} + 99 \, a d e^{4}\right )} x^{2} -{\left (4 \, c d^{4} e - 297 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{693 \, e^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*c*e^5*x^5 + 161*c*d*e^4*x^4 + 8*c*d^5 + 99*a*d^3*e^2 + (113*c*d^2*e^3 + 99*a*e^5)*x^3 + 3*(c*d^3*e^2
+ 99*a*d*e^4)*x^2 - (4*c*d^4*e - 297*a*d^2*e^3)*x)*sqrt(e*x + d)/e^3

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Sympy [A]  time = 3.19168, size = 218, normalized size = 3.46 \begin{align*} \begin{cases} \frac{2 a d^{3} \sqrt{d + e x}}{7 e} + \frac{6 a d^{2} x \sqrt{d + e x}}{7} + \frac{6 a d e x^{2} \sqrt{d + e x}}{7} + \frac{2 a e^{2} x^{3} \sqrt{d + e x}}{7} + \frac{16 c d^{5} \sqrt{d + e x}}{693 e^{3}} - \frac{8 c d^{4} x \sqrt{d + e x}}{693 e^{2}} + \frac{2 c d^{3} x^{2} \sqrt{d + e x}}{231 e} + \frac{226 c d^{2} x^{3} \sqrt{d + e x}}{693} + \frac{46 c d e x^{4} \sqrt{d + e x}}{99} + \frac{2 c e^{2} x^{5} \sqrt{d + e x}}{11} & \text{for}\: e \neq 0 \\d^{\frac{5}{2}} \left (a x + \frac{c x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*a*d**3*sqrt(d + e*x)/(7*e) + 6*a*d**2*x*sqrt(d + e*x)/7 + 6*a*d*e*x**2*sqrt(d + e*x)/7 + 2*a*e**2
*x**3*sqrt(d + e*x)/7 + 16*c*d**5*sqrt(d + e*x)/(693*e**3) - 8*c*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*c*d**3*x*
*2*sqrt(d + e*x)/(231*e) + 226*c*d**2*x**3*sqrt(d + e*x)/693 + 46*c*d*e*x**4*sqrt(d + e*x)/99 + 2*c*e**2*x**5*
sqrt(d + e*x)/11, Ne(e, 0)), (d**(5/2)*(a*x + c*x**3/3), True))

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Giac [B]  time = 1.40527, size = 331, normalized size = 5.25 \begin{align*} \frac{2}{3465} \,{\left (33 \,{\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 d^{2} e^{\left (-2\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2} + 22 \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} c d e^{\left (-2\right )} + 462 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d +{\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 e^{\left (-2\right )} + 33 \,{\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\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3465*(33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*c*d^2*e^(-2) + 1155*(x*e + d)^
(3/2)*a*d^2 + 22*(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)*c*d*e^(-2) + 462*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*a*d + (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*e^(-2) + 33*(15*(x*e
+ d)^(7/2) - 42*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*a)*e^(-1)