3.1431 \(\int \frac{(A+B x) \left (a+c x^2\right )}{\sqrt{d+e x}} \, dx\)

Optimal. Leaf size=114 \[ \frac{2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac{2 c (d+e x)^{5/2} (3 B d-A e)}{5 e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^4 + (2*(3*B*c*d^2 - 2*A*c*d*e +
a*B*e^2)*(d + e*x)^(3/2))/(3*e^4) - (2*c*(3*B*d - A*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 = 0.135579, antiderivative size = 114, 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 \[ \frac{2 (d+e x)^{3/2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{3 e^4}-\frac{2 \sqrt{d+e x} \left (a e^2+c d^2\right ) (B d-A e)}{e^4}-\frac{2 c (d+e x)^{5/2} (3 B d-A e)}{5 e^4}+\frac{2 B c (d+e x)^{7/2}}{7 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(B*d - A*e)*(c*d^2 + a*e^2)*Sqrt[d + e*x])/e^4 + (2*(3*B*c*d^2 - 2*A*c*d*e +
a*B*e^2)*(d + e*x)^(3/2))/(3*e^4) - (2*c*(3*B*d - A*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 = 23.8659, size = 112, normalized size = 0.98 \[ \frac{2 B c \left (d + e x\right )^{\frac{7}{2}}}{7 e^{4}} + \frac{2 c \left (d + e x\right )^{\frac{5}{2}} \left (A e - 3 B d\right )}{5 e^{4}} + \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{e^{4}} \]

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

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Mathematica [A]  time = 0.102008, size = 96, normalized size = 0.84 \[ \frac{2 \sqrt{d+e x} \left (105 a A e^3+35 a B e^2 (e x-2 d)+7 A c e \left (8 d^2-4 d e x+3 e^2 x^2\right )-3 B c \left (16 d^3-8 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )}{105 e^4} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d + e*x]*(105*a*A*e^3 + 35*a*B*e^2*(-2*d + e*x) + 7*A*c*e*(8*d^2 - 4*d*e
*x + 3*e^2*x^2) - 3*B*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 = 0.01, size = 101, normalized size = 0.9 \[{\frac{30\,Bc{x}^{3}{e}^{3}+42\,Ac{e}^{3}{x}^{2}-36\,Bcd{e}^{2}{x}^{2}-56\,Acd{e}^{2}x+70\,Ba{e}^{3}x+48\,Bc{d}^{2}ex+210\,aA{e}^{3}+112\,Ac{d}^{2}e-140\,aBd{e}^{2}-96\,Bc{d}^{3}}{105\,{e}^{4}}\sqrt{ex+d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.683398, size = 140, normalized size = 1.23 \[ \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} B c - 21 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 105 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} \sqrt{e x + d}\right )}}{105 \, e^{4}} \]

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]

2/105*(15*(e*x + d)^(7/2)*B*c - 21*(3*B*c*d - A*c*e)*(e*x + d)^(5/2) + 35*(3*B*c
*d^2 - 2*A*c*d*e + B*a*e^2)*(e*x + d)^(3/2) - 105*(B*c*d^3 - A*c*d^2*e + B*a*d*e
^2 - A*a*e^3)*sqrt(e*x + d))/e^4

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Fricas [A]  time = 0.266455, size = 135, normalized size = 1.18 \[ \frac{2 \,{\left (15 \, B c e^{3} x^{3} - 48 \, B c d^{3} + 56 \, A c d^{2} e - 70 \, B a d e^{2} + 105 \, A a e^{3} - 3 \,{\left (6 \, B c d e^{2} - 7 \, A c e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e - 28 \, A c d e^{2} + 35 \, B a e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)*(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 + 56*A*c*d^2*e - 70*B*a*d*e^2 + 105*A*a*e^3 -
3*(6*B*c*d*e^2 - 7*A*c*e^3)*x^2 + (24*B*c*d^2*e - 28*A*c*d*e^2 + 35*B*a*e^3)*x)
*sqrt(e*x + d)/e^4

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Sympy [A]  time = 23.0064, size = 374, normalized size = 3.28 \[ \begin{cases} - \frac{\frac{2 A a d}{\sqrt{d + e x}} + 2 A a \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{2 A c d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 A c \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{2 B a d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{2 B a \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 B c d \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{3}} + \frac{2 B c \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4}}{\sqrt{d}} & \text{otherwise} \end{cases} \]

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]

Piecewise((-(2*A*a*d/sqrt(d + e*x) + 2*A*a*(-d/sqrt(d + e*x) - sqrt(d + e*x)) +
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*a*d*(-d/sqrt(d + e*x) - sqrt(d + e*x))/e + 2*B*a*(d**2/sqrt
(d + e*x) + 2*d*sqrt(d + e*x) - (d + e*x)**(3/2)/3)/e + 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, Ne(e, 0)), ((A*a*x + A*c*x**3/3
+ B*a*x**2/2 + B*c*x**4/4)/sqrt(d), True))

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GIAC/XCAS [A]  time = 0.285066, size = 205, normalized size = 1.8 \[ \frac{2}{105} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} B a e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} e^{8} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d e^{8} + 15 \, \sqrt{x e + d} d^{2} e^{8}\right )} A c e^{\left (-10\right )} + 3 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} e^{18} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} e^{18} - 35 \, \sqrt{x e + d} d^{3} e^{18}\right )} B c e^{\left (-21\right )} + 105 \, \sqrt{x e + d} A a\right )} e^{\left (-1\right )} \]

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]

2/105*(35*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a*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)*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) + 105*sqrt(x*e + d)*A*a)*e^(-1)