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

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 = 0.139229, 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 \[ \frac{2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac{2 (d+e x)^{3/2} \left (a e^2+c d^2\right ) (B d-A e)}{3 e^4}-\frac{2 c (d+e x)^{7/2} (3 B d-A e)}{7 e^4}+\frac{2 B c (d+e x)^{9/2}}{9 e^4} \]

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)

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Rubi in Sympy [A]  time = 24.0428, size = 114, normalized size = 0.98 \[ \frac{2 B c \left (d + e x\right )^{\frac{9}{2}}}{9 e^{4}} + \frac{2 c \left (d + e x\right )^{\frac{7}{2}} \left (A e - 3 B d\right )}{7 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{3 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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**4) + 2*c*(d + e*x)**(7/2)*(A*e - 3*B*d)/(7*e**4) +
2*(d + e*x)**(5/2)*(-2*A*c*d*e + B*a*e**2 + 3*B*c*d**2)/(5*e**4) + 2*(d + e*x)**
(3/2)*(A*e - B*d)*(a*e**2 + c*d**2)/(3*e**4)

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Mathematica [A]  time = 0.145948, size = 96, normalized size = 0.83 \[ \frac{2 (d+e x)^{3/2} \left (105 a A e^3+21 a B e^2 (3 e x-2 d)+3 A c e \left (8 d^2-12 d e x+15 e^2 x^2\right )+B c \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4} \]

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 - 1
2*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 = 0.007, size = 101, normalized size = 0.9 \[{\frac{70\,Bc{x}^{3}{e}^{3}+90\,Ac{e}^{3}{x}^{2}-60\,Bcd{e}^{2}{x}^{2}-72\,Acd{e}^{2}x+126\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+210\,aA{e}^{3}+48\,Ac{d}^{2}e-84\,aBd{e}^{2}-32\,Bc{d}^{3}}{315\,{e}^{4}} \left ( ex+d \right ) ^{{\frac{3}{2}}}} \]

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 = 0.684234, size = 140, normalized size = 1.21 \[ \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} B c - 45 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*sqrt(e*x + d),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 = 0.266051, size = 193, normalized size = 1.66 \[ \frac{2 \,{\left (35 \, B c e^{4} x^{4} - 16 \, B c d^{4} + 24 \, A c d^{3} e - 42 \, B a d^{2} e^{2} + 105 \, A a d e^{3} + 5 \,{\left (B c d e^{3} + 9 \, A c e^{4}\right )} x^{3} - 3 \,{\left (2 \, B c d^{2} e^{2} - 3 \, A c d e^{3} - 21 \, B a e^{4}\right )} x^{2} +{\left (8 \, B c d^{3} e - 12 \, A c d^{2} e^{2} + 21 \, B a d e^{3} + 105 \, A a e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*sqrt(e*x + d),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 = 4.61945, size = 131, normalized size = 1.13 \[ \frac{2 \left (\frac{B c \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (A c e - 3 B c d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (A a e^{3} + A c d^{2} e - B a d e^{2} - B c d^{3}\right )}{3 e^{3}}\right )}{e} \]

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/XCAS [A]  time = 0.288176, size = 207, normalized size = 1.78 \[ \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} B a e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{12} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{12} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{12}\right )} A c e^{\left (-14\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} e^{24} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d e^{24} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} e^{24} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} e^{24}\right )} B c e^{\left (-27\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} A a\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(B*x + A)*sqrt(e*x + d),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)*e^12 - 42*(x*e + d)^(5/2)*d*e^12 + 35*(x*e + d)^(3/2)*d^2*e^12)*A*c*e^(-1
4) + (35*(x*e + d)^(9/2)*e^24 - 135*(x*e + d)^(7/2)*d*e^24 + 189*(x*e + d)^(5/2)
*d^2*e^24 - 105*(x*e + d)^(3/2)*d^3*e^24)*B*c*e^(-27) + 105*(x*e + d)^(3/2)*A*a)
*e^(-1)