3.1429 \(\int (A+B x) (d+e x)^{3/2} (a+c x^2) \, dx\)

Optimal. Leaf size=116 \[ \frac{2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{2 c (d+e x)^{9/2} (3 B d-A e)}{9 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]

[Out]

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

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Rubi [A]  time = 0.0616693, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ \frac{2 (d+e x)^{7/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right ) (B d-A e)}{5 e^4}-\frac{2 c (d+e x)^{9/2} (3 B d-A e)}{9 e^4}+\frac{2 B c (d+e x)^{11/2}}{11 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(3/2)*(a + c*x^2),x]

[Out]

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

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0987872, size = 99, normalized size = 0.85 \[ \frac{2 (d+e x)^{5/2} \left (11 A e \left (63 a e^2+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )-3 B \left (33 a e^2 (2 d-5 e x)+c \left (-40 d^2 e x+16 d^3+70 d e^2 x^2-105 e^3 x^3\right )\right )\right )}{3465 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(3/2)*(a + c*x^2),x]

[Out]

(2*(d + e*x)^(5/2)*(11*A*e*(63*a*e^2 + c*(8*d^2 - 20*d*e*x + 35*e^2*x^2)) - 3*B*(33*a*e^2*(2*d - 5*e*x) + c*(1
6*d^3 - 40*d^2*e*x + 70*d*e^2*x^2 - 105*e^3*x^3))))/(3465*e^4)

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Maple [A]  time = 0.006, size = 101, normalized size = 0.9 \begin{align*}{\frac{630\,Bc{x}^{3}{e}^{3}+770\,Ac{e}^{3}{x}^{2}-420\,Bcd{e}^{2}{x}^{2}-440\,Acd{e}^{2}x+990\,Ba{e}^{3}x+240\,Bc{d}^{2}ex+1386\,aA{e}^{3}+176\,Ac{d}^{2}e-396\,aBd{e}^{2}-96\,Bc{d}^{3}}{3465\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)*(c*x^2+a),x)

[Out]

2/3465*(e*x+d)^(5/2)*(315*B*c*e^3*x^3+385*A*c*e^3*x^2-210*B*c*d*e^2*x^2-220*A*c*d*e^2*x+495*B*a*e^3*x+120*B*c*
d^2*e*x+693*A*a*e^3+88*A*c*d^2*e-198*B*a*d*e^2-48*B*c*d^3)/e^4

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Maxima [A]  time = 1.03168, size = 140, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (315 \,{\left (e x + d\right )}^{\frac{11}{2}} B c - 385 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{9}{2}} + 495 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} - 693 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{3465 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+a),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*c - 385*(3*B*c*d - A*c*e)*(e*x + d)^(9/2) + 495*(3*B*c*d^2 - 2*A*c*d*e + B*a*e^
2)*(e*x + d)^(7/2) - 693*(B*c*d^3 - A*c*d^2*e + B*a*d*e^2 - A*a*e^3)*(e*x + d)^(5/2))/e^4

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Fricas [A]  time = 1.42335, size = 458, normalized size = 3.95 \begin{align*} \frac{2 \,{\left (315 \, B c e^{5} x^{5} - 48 \, B c d^{5} + 88 \, A c d^{4} e - 198 \, B a d^{3} e^{2} + 693 \, A a d^{2} e^{3} + 35 \,{\left (12 \, B c d e^{4} + 11 \, A c e^{5}\right )} x^{4} + 5 \,{\left (3 \, B c d^{2} e^{3} + 110 \, A c d e^{4} + 99 \, B a e^{5}\right )} x^{3} - 3 \,{\left (6 \, B c d^{3} e^{2} - 11 \, A c d^{2} e^{3} - 264 \, B a d e^{4} - 231 \, A a e^{5}\right )} x^{2} +{\left (24 \, B c d^{4} e - 44 \, A c d^{3} e^{2} + 99 \, B a d^{2} e^{3} + 1386 \, A a d e^{4}\right )} x\right )} \sqrt{e x + d}}{3465 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+a),x, algorithm="fricas")

[Out]

2/3465*(315*B*c*e^5*x^5 - 48*B*c*d^5 + 88*A*c*d^4*e - 198*B*a*d^3*e^2 + 693*A*a*d^2*e^3 + 35*(12*B*c*d*e^4 + 1
1*A*c*e^5)*x^4 + 5*(3*B*c*d^2*e^3 + 110*A*c*d*e^4 + 99*B*a*e^5)*x^3 - 3*(6*B*c*d^3*e^2 - 11*A*c*d^2*e^3 - 264*
B*a*d*e^4 - 231*A*a*e^5)*x^2 + (24*B*c*d^4*e - 44*A*c*d^3*e^2 + 99*B*a*d^2*e^3 + 1386*A*a*d*e^4)*x)*sqrt(e*x +
d)/e^4

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Sympy [A]  time = 11.8672, size = 379, normalized size = 3.27 \begin{align*} A a d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 A a \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{2 A c d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 A c \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} + \frac{2 B a d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 B c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{4}} + \frac{2 B c \left (\frac{d^{4} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{4 d^{3} \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{6 d^{2} \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{4 d \left (d + e x\right )^{\frac{9}{2}}}{9} + \frac{\left (d + e x\right )^{\frac{11}{2}}}{11}\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(3/2)*(c*x**2+a),x)

[Out]

A*a*d*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 2*A*a*(-d*(d + e*x)**(3/2)/3 + (d +
e*x)**(5/2)/5)/e + 2*A*c*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 2*A
*c*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 +
2*B*a*d*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 2*B*a*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(
5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 2*B*c*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e
*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**4 + 2*B*c*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*
(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4

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Giac [B]  time = 1.14264, size = 447, normalized size = 3.85 \begin{align*} \frac{2}{3465} \,{\left (231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a d e^{\left (-1\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 c d e^{\left (-2\right )} + 11 \,{\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 )} B c d e^{\left (-3\right )} + 1155 \,{\left (x e + d\right )}^{\frac{3}{2}} A a d + 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 )} B a e^{\left (-1\right )} + 11 \,{\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 )} 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 )} B c e^{\left (-3\right )} + 231 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} A a\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(c*x^2+a),x, algorithm="giac")

[Out]

2/3465*(231*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*d*e^(-1) + 33*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/
2)*d + 35*(x*e + d)^(3/2)*d^2)*A*c*d*e^(-2) + 11*(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)*B*c*d*e^(-3) + 1155*(x*e + d)^(3/2)*A*a*d + 33*(15*(x*e + d)^(7/2) - 42*(x
*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2)*B*a*e^(-1) + 11*(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)*A*c*e^(-2) + (315*(x*e + d)^(11/2) - 1540*(x*e + d)^(9/2)*d + 297
0*(x*e + d)^(7/2)*d^2 - 2772*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4)*B*c*e^(-3) + 231*(3*(x*e + d)^(5/
2) - 5*(x*e + d)^(3/2)*d)*A*a)*e^(-1)