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

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 116, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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 = 0.07, size = 96, normalized size = 0.83 \begin {gather*} \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 (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \end {gather*}

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)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 117, normalized size = 1.01 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (105 a A e^3+63 a B e^2 (d+e x)-105 a B d e^2+105 A c d^2 e-126 A c d e (d+e x)+45 A c e (d+e x)^2-105 B c d^3+189 B c d^2 (d+e x)-135 B c d (d+e x)^2+35 B c (d+e x)^3\right )}{315 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(-105*B*c*d^3 + 105*A*c*d^2*e - 105*a*B*d*e^2 + 105*a*A*e^3 + 189*B*c*d^2*(d + e*x) - 126*A
*c*d*e*(d + e*x) + 63*a*B*e^2*(d + e*x) - 135*B*c*d*(d + e*x)^2 + 45*A*c*e*(d + e*x)^2 + 35*B*c*(d + e*x)^3))/
(315*e^4)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 143, normalized size = 1.23 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

giac [B]  time = 0.16, size = 327, normalized size = 2.82 \begin {gather*} \frac {2}{315} \, {\left (105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} B a d e^{\left (-1\right )} + 21 \, {\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 )} A c d e^{\left (-2\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} B c d e^{\left (-3\right )} + 21 \, {\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 )} B a e^{\left (-1\right )} + 9 \, {\left (5 \, {\left (x e + d\right )}^{\frac {7}{2}} - 21 \, {\left (x e + d\right )}^{\frac {5}{2}} d + 35 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {x e + d} d^{3}\right )} A c e^{\left (-2\right )} + {\left (35 \, {\left (x e + d\right )}^{\frac {9}{2}} - 180 \, {\left (x e + d\right )}^{\frac {7}{2}} d + 378 \, {\left (x e + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (x e + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {x e + d} d^{4}\right )} B c e^{\left (-3\right )} + 315 \, \sqrt {x e + d} A a d + 105 \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {x e + d} d\right )} A a\right )} e^{\left (-1\right )} \end {gather*}

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*(105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*d*e^(-1) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*A*c*d*e^(-2) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 3
5*sqrt(x*e + d)*d^3)*B*c*d*e^(-3) + 21*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*e
^(-1) + 9*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*c*e^(-2
) + (35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt
(x*e + d)*d^4)*B*c*e^(-3) + 315*sqrt(x*e + d)*A*a*d + 105*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a)*e^(-1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 101, normalized size = 0.87 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (35 B c \,x^{3} e^{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 a B d \,e^{2}-16 B c \,d^{3}\right )}{315 e^{4}} \end {gather*}

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

________________________________________________________________________________________

maxima [A]  time = 0.77, size = 104, normalized size = 0.90 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

mupad [B]  time = 1.74, size = 100, normalized size = 0.86 \begin {gather*} \frac {{\left (d+e\,x\right )}^{5/2}\,\left (6\,B\,c\,d^2-4\,A\,c\,d\,e+2\,B\,a\,e^2\right )}{5\,e^4}+\frac {2\,B\,c\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {2\,c\,\left (A\,e-3\,B\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4}+\frac {2\,\left (c\,d^2+a\,e^2\right )\,\left (A\,e-B\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((d + e*x)^(5/2)*(2*B*a*e^2 + 6*B*c*d^2 - 4*A*c*d*e))/(5*e^4) + (2*B*c*(d + e*x)^(9/2))/(9*e^4) + (2*c*(A*e -
3*B*d)*(d + e*x)^(7/2))/(7*e^4) + (2*(a*e^2 + c*d^2)*(A*e - B*d)*(d + e*x)^(3/2))/(3*e^4)

________________________________________________________________________________________

sympy [A]  time = 3.74, size = 131, normalized size = 1.13 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________