3.1217 \(\int \frac{(A+B x) \left (b x+c x^2\right )}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=124 \[ -\frac{2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac{2 d \sqrt{d+e x} (B d-A e) (c d-b e)}{e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]

[Out]

(-2*d*(B*d - A*e)*(c*d - b*e)*Sqrt[d + e*x])/e^4 + (2*(B*d*(3*c*d - 2*b*e) - A*e
*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x
)^(5/2))/(5*e^4) + (2*B*c*(d + e*x)^(7/2))/(7*e^4)

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Rubi [A]  time = 0.206323, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{2 (d+e x)^{5/2} (-A c e-b B e+3 B c d)}{5 e^4}+\frac{2 (d+e x)^{3/2} (B d (3 c d-2 b e)-A e (2 c d-b e))}{3 e^4}-\frac{2 d \sqrt{d+e x} (B d-A e) (c d-b e)}{e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d*(B*d - A*e)*(c*d - b*e)*Sqrt[d + e*x])/e^4 + (2*(B*d*(3*c*d - 2*b*e) - A*e
*(2*c*d - b*e))*(d + e*x)^(3/2))/(3*e^4) - (2*(3*B*c*d - b*B*e - A*c*e)*(d + e*x
)^(5/2))/(5*e^4) + (2*B*c*(d + e*x)^(7/2))/(7*e^4)

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Rubi in Sympy [A]  time = 33.0833, size = 126, normalized size = 1.02 \[ \frac{2 B c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} - \frac{2 d \sqrt{d + e x} \left (A e - B d\right ) \left (b e - c d\right )}{e^{4}} + \frac{2 \left (d + e x\right )^{\frac{5}{2}} \left (A c e + B b e - 3 B c d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (A b e^{2} - 2 A c d e - 2 B b d e + 3 B c d^{2}\right )}{3 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**(1/2),x)

[Out]

2*B*c*(d + e*x)**(7/2)/(7*e**4) - 2*d*sqrt(d + e*x)*(A*e - B*d)*(b*e - c*d)/e**4
+ 2*(d + e*x)**(5/2)*(A*c*e + B*b*e - 3*B*c*d)/(5*e**4) + 2*(d + e*x)**(3/2)*(A
*b*e**2 - 2*A*c*d*e - 2*B*b*d*e + 3*B*c*d**2)/(3*e**4)

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Mathematica [A]  time = 0.163168, size = 113, normalized size = 0.91 \[ \frac{2 \sqrt{d+e x} \left (7 A e \left (5 b e (e x-2 d)+c \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )+B \left (7 b e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 c \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )\right )}{105 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(7*A*e*(5*b*e*(-2*d + e*x) + c*(8*d^2 - 4*d*e*x + 3*e^2*x^2)) +
B*(7*b*e*(8*d^2 - 4*d*e*x + 3*e^2*x^2) - 3*c*(16*d^3 - 8*d^2*e*x + 6*d*e^2*x^2
- 5*e^3*x^3))))/(105*e^4)

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Maple [A]  time = 0.006, size = 121, normalized size = 1. \[ -{\frac{-30\,Bc{x}^{3}{e}^{3}-42\,Ac{e}^{3}{x}^{2}-42\,Bb{e}^{3}{x}^{2}+36\,Bcd{e}^{2}{x}^{2}-70\,Ab{e}^{3}x+56\,Acd{e}^{2}x+56\,Bbd{e}^{2}x-48\,Bc{d}^{2}ex+140\,Abd{e}^{2}-112\,Ac{d}^{2}e-112\,Bb{d}^{2}e+96\,Bc{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)/(e*x+d)^(1/2),x)

[Out]

-2/105*(-15*B*c*e^3*x^3-21*A*c*e^3*x^2-21*B*b*e^3*x^2+18*B*c*d*e^2*x^2-35*A*b*e^
3*x+28*A*c*d*e^2*x+28*B*b*d*e^2*x-24*B*c*d^2*e*x+70*A*b*d*e^2-56*A*c*d^2*e-56*B*
b*d^2*e+48*B*c*d^3)*(e*x+d)^(1/2)/e^4

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Maxima [A]  time = 0.70534, size = 151, normalized size = 1.22 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c - 21 \,{\left (3 \, B c d -{\left (B b + A c\right )} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B c d^{2} + A b e^{2} - 2 \,{\left (B b + A c\right )} d e\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B c d^{3} + A b d e^{2} -{\left (B b + A c\right )} d^{2} e\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/sqrt(e*x + d),x, algorithm="maxima")

[Out]

2/105*(15*(e*x + d)^(7/2)*B*c - 21*(3*B*c*d - (B*b + A*c)*e)*(e*x + d)^(5/2) + 3
5*(3*B*c*d^2 + A*b*e^2 - 2*(B*b + A*c)*d*e)*(e*x + d)^(3/2) - 105*(B*c*d^3 + A*b
*d*e^2 - (B*b + A*c)*d^2*e)*sqrt(e*x + d))/e^4

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Fricas [A]  time = 0.314798, size = 146, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} - 70 \, A b d e^{2} + 56 \,{\left (B b + A c\right )} d^{2} e - 3 \,{\left (6 \, B c d e^{2} - 7 \,{\left (B b + A c\right )} e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e + 35 \, A b e^{3} - 28 \,{\left (B b + A c\right )} d e^{2}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/sqrt(e*x + d),x, algorithm="fricas")

[Out]

2/105*(15*B*c*e^3*x^3 - 48*B*c*d^3 - 70*A*b*d*e^2 + 56*(B*b + A*c)*d^2*e - 3*(6*
B*c*d*e^2 - 7*(B*b + A*c)*e^3)*x^2 + (24*B*c*d^2*e + 35*A*b*e^3 - 28*(B*b + A*c)
*d*e^2)*x)*sqrt(e*x + d)/e^4

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Sympy [A]  time = 29.2106, size = 430, normalized size = 3.47 \[ \begin{cases} - \frac{\frac{2 A b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 A b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 A c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B b d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 B b \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B c \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{\frac{A b x^{2}}{2} + \frac{B c x^{4}}{4} + \frac{x^{3} \left (A c + B b\right )}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)/(e*x+d)**(1/2),x)

[Out]

Piecewise((-(2*A*b*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*A*b*(d**2/sqrt(d +
e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 2*A*c*d*(d**2/sqrt(d + e*x)
+ 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*A*c*(-d**3/sqrt(d + e*x) - 3*
d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e**2 + 2*B*b*d*(d*
*2/sqrt(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e**2 + 2*B*b*(-d**3/s
qrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) - (d + e*x)**(5/2)/5)/e
**2 + 2*B*c*d*(-d**3/sqrt(d + e*x) - 3*d**2*sqrt(d + e*x) + d*(d + e*x)**(3/2) -
(d + e*x)**(5/2)/5)/e**3 + 2*B*c*(d**4/sqrt(d + e*x) + 4*d**3*sqrt(d + e*x) - 2
*d**2*(d + e*x)**(3/2) + 4*d*(d + e*x)**(5/2)/5 - (d + e*x)**(7/2)/7)/e**3)/e, N
e(e, 0)), ((A*b*x**2/2 + B*c*x**4/4 + x**3*(A*c + B*b)/3)/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.27934, size = 252, normalized size = 2.03 \[ \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} A b e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} B b e^{\left (-10\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A c e^{\left (-10\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B c e^{\left (-21\right )}\right )} e^{\left (-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)*(B*x + A)/sqrt(e*x + d),x, algorithm="giac")

[Out]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*b*e^(-1) + 7*(3*(x*e + d)^(5/2
)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*B*b*e^(-10) + 7*(3*
(x*e + d)^(5/2)*e^8 - 10*(x*e + d)^(3/2)*d*e^8 + 15*sqrt(x*e + d)*d^2*e^8)*A*c*e
^(-10) + 3*(5*(x*e + d)^(7/2)*e^18 - 21*(x*e + d)^(5/2)*d*e^18 + 35*(x*e + d)^(3
/2)*d^2*e^18 - 35*sqrt(x*e + d)*d^3*e^18)*B*c*e^(-21))*e^(-1)