3.1430 \(\int (A+B x) \sqrt{d+e x} (a+c x^2) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0511429, 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} \[ \frac{2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4}-\frac{2 c (d+e x)^{7/2} (3 B d-A e)}{7 e^4}+\frac{2 B c (d+e x)^{9/2}}{9 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0740469, size = 96, normalized size = 0.83 \[ \frac{2 (d+e x)^{3/2} \left (105 a A e^3+21 a B e^2 (3 e x-2 d)+3 A c e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B c \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(105*a*A*e^3 + 21*a*B*e^2*(-2*d + 3*e*x) + 3*A*c*e*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + B*c*(-
16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3)))/(315*e^4)

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Maple [A]  time = 0.006, size = 101, normalized size = 0.9 \begin{align*}{\frac{70\,Bc{x}^{3}{e}^{3}+90\,Ac{e}^{3}{x}^{2}-60\,Bcd{e}^{2}{x}^{2}-72\,Acd{e}^{2}x+126\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+210\,aA{e}^{3}+48\,Ac{d}^{2}e-84\,aBd{e}^{2}-32\,Bc{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(e*x+d)^(3/2)*(35*B*c*e^3*x^3+45*A*c*e^3*x^2-30*B*c*d*e^2*x^2-36*A*c*d*e^2*x+63*B*a*e^3*x+24*B*c*d^2*e*x
+105*A*a*e^3+24*A*c*d^2*e-42*B*a*d*e^2-16*B*c*d^3)/e^4

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Maxima [A]  time = 1.00152, size = 140, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c - 45 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\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{3}{2}}\right )}}{315 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*(e*x + d)^(9/2)*B*c - 45*(3*B*c*d - A*c*e)*(e*x + d)^(7/2) + 63*(3*B*c*d^2 - 2*A*c*d*e + B*a*e^2)*(e
*x + d)^(5/2) - 105*(B*c*d^3 - A*c*d^2*e + B*a*d*e^2 - A*a*e^3)*(e*x + d)^(3/2))/e^4

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Fricas [A]  time = 1.6967, size = 338, normalized size = 2.91 \begin{align*} \frac{2 \,{\left (35 \, B c e^{4} x^{4} - 16 \, B c d^{4} + 24 \, A c d^{3} e - 42 \, B a d^{2} e^{2} + 105 \, A a d e^{3} + 5 \,{\left (B c d e^{3} + 9 \, A c e^{4}\right )} x^{3} - 3 \,{\left (2 \, B c d^{2} e^{2} - 3 \, A c d e^{3} - 21 \, B a e^{4}\right )} x^{2} +{\left (8 \, B c d^{3} e - 12 \, A c d^{2} e^{2} + 21 \, B a d e^{3} + 105 \, A a e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(35*B*c*e^4*x^4 - 16*B*c*d^4 + 24*A*c*d^3*e - 42*B*a*d^2*e^2 + 105*A*a*d*e^3 + 5*(B*c*d*e^3 + 9*A*c*e^4)
*x^3 - 3*(2*B*c*d^2*e^2 - 3*A*c*d*e^3 - 21*B*a*e^4)*x^2 + (8*B*c*d^3*e - 12*A*c*d^2*e^2 + 21*B*a*d*e^3 + 105*A
*a*e^4)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 2.83482, size = 131, normalized size = 1.13 \begin{align*} \frac{2 \left (\frac{B c \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A c e - 3 B c d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a e^{3} + A c d^{2} e - B a d e^{2} - B c d^{3}\right )}{3 e^{3}}\right )}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14616, size = 188, normalized size = 1.62 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a e^{\left (-1\right )} + 3 \,{\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 e^{\left (-2\right )} +{\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 e^{\left (-3\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a\right )} e^{\left (-1\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(21*(3*(x*e + d)^(5/2) - 5*(x*e + d)^(3/2)*d)*B*a*e^(-1) + 3*(15*(x*e + d)^(7/2) - 42*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2)*A*c*e^(-2) + (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*e^(-3) + 105*(x*e + d)^(3/2)*A*a)*e^(-1)