3.1432 \(\int \frac{(A+B x) \left (a+c x^2\right )}{(d+e x)^{3/2}} \, dx\)
Optimal. Leaf size=112 \[ \frac{2 \sqrt{d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
[Out]
(2*(B*d - A*e)*(c*d^2 + a*e^2))/(e^4*Sqrt[d + e*x]) + (2*(3*B*c*d^2 - 2*A*c*d*e
+ a*B*e^2)*Sqrt[d + e*x])/e^4 - (2*c*(3*B*d - A*e)*(d + e*x)^(3/2))/(3*e^4) + (2
*B*c*(d + e*x)^(5/2))/(5*e^4)
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Rubi [A] time = 0.136161, antiderivative size = 112, 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
\[ \frac{2 \sqrt{d+e x} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4}+\frac{2 \left (a e^2+c d^2\right ) (B d-A e)}{e^4 \sqrt{d+e x}}-\frac{2 c (d+e x)^{3/2} (3 B d-A e)}{3 e^4}+\frac{2 B c (d+e x)^{5/2}}{5 e^4} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]
[Out]
(2*(B*d - A*e)*(c*d^2 + a*e^2))/(e^4*Sqrt[d + e*x]) + (2*(3*B*c*d^2 - 2*A*c*d*e
+ a*B*e^2)*Sqrt[d + e*x])/e^4 - (2*c*(3*B*d - A*e)*(d + e*x)^(3/2))/(3*e^4) + (2
*B*c*(d + e*x)^(5/2))/(5*e^4)
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Rubi in Sympy [A] time = 23.8442, size = 110, normalized size = 0.98 \[ \frac{2 B c \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{2 c \left (d + e x\right )^{\frac{3}{2}} \left (A e - 3 B d\right )}{3 e^{4}} + \frac{2 \sqrt{d + e x} \left (- 2 A c d e + B a e^{2} + 3 B c d^{2}\right )}{e^{4}} - \frac{2 \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )}{e^{4} \sqrt{d + e x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(3/2),x)
[Out]
2*B*c*(d + e*x)**(5/2)/(5*e**4) + 2*c*(d + e*x)**(3/2)*(A*e - 3*B*d)/(3*e**4) +
2*sqrt(d + e*x)*(-2*A*c*d*e + B*a*e**2 + 3*B*c*d**2)/e**4 - 2*(A*e - B*d)*(a*e**
2 + c*d**2)/(e**4*sqrt(d + e*x))
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Mathematica [A] time = 0.118564, size = 97, normalized size = 0.87 \[ \frac{6 B \left (5 a e^2 (2 d+e x)+c \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )-10 A e \left (3 a e^2+c \left (8 d^2+4 d e x-e^2 x^2\right )\right )}{15 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^(3/2),x]
[Out]
(-10*A*e*(3*a*e^2 + c*(8*d^2 + 4*d*e*x - e^2*x^2)) + 6*B*(5*a*e^2*(2*d + e*x) +
c*(16*d^3 + 8*d^2*e*x - 2*d*e^2*x^2 + e^3*x^3)))/(15*e^4*Sqrt[d + e*x])
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Maple [A] time = 0.006, size = 101, normalized size = 0.9 \[ -{\frac{-6\,Bc{x}^{3}{e}^{3}-10\,Ac{e}^{3}{x}^{2}+12\,Bcd{e}^{2}{x}^{2}+40\,Acd{e}^{2}x-30\,Ba{e}^{3}x-48\,Bc{d}^{2}ex+30\,aA{e}^{3}+80\,Ac{d}^{2}e-60\,aBd{e}^{2}-96\,Bc{d}^{3}}{15\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+a)/(e*x+d)^(3/2),x)
[Out]
-2/15/(e*x+d)^(1/2)*(-3*B*c*e^3*x^3-5*A*c*e^3*x^2+6*B*c*d*e^2*x^2+20*A*c*d*e^2*x
-15*B*a*e^3*x-24*B*c*d^2*e*x+15*A*a*e^3+40*A*c*d^2*e-30*B*a*d*e^2-48*B*c*d^3)/e^
4
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Maxima [A] time = 0.685363, size = 151, normalized size = 1.35 \[ \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} B c - 5 \,{\left (3 \, B c d - A c e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (3 \, B c d^{2} - 2 \, A c d e + B a e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{15 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )}}{\sqrt{e x + d} e^{3}}\right )}}{15 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="maxima")
[Out]
2/15*((3*(e*x + d)^(5/2)*B*c - 5*(3*B*c*d - A*c*e)*(e*x + d)^(3/2) + 15*(3*B*c*d
^2 - 2*A*c*d*e + B*a*e^2)*sqrt(e*x + d))/e^3 + 15*(B*c*d^3 - A*c*d^2*e + B*a*d*e
^2 - A*a*e^3)/(sqrt(e*x + d)*e^3))/e
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Fricas [A] time = 0.264841, size = 135, normalized size = 1.21 \[ \frac{2 \,{\left (3 \, B c e^{3} x^{3} + 48 \, B c d^{3} - 40 \, A c d^{2} e + 30 \, B a d e^{2} - 15 \, A a e^{3} -{\left (6 \, B c d e^{2} - 5 \, A c e^{3}\right )} x^{2} +{\left (24 \, B c d^{2} e - 20 \, A c d e^{2} + 15 \, B a e^{3}\right )} x\right )}}{15 \, \sqrt{e x + d} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="fricas")
[Out]
2/15*(3*B*c*e^3*x^3 + 48*B*c*d^3 - 40*A*c*d^2*e + 30*B*a*d*e^2 - 15*A*a*e^3 - (6
*B*c*d*e^2 - 5*A*c*e^3)*x^2 + (24*B*c*d^2*e - 20*A*c*d*e^2 + 15*B*a*e^3)*x)/(sqr
t(e*x + d)*e^4)
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Sympy [A] time = 19.5986, size = 2139, normalized size = 19.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+a)/(e*x+d)**(3/2),x)
[Out]
-2*A*a/(e*sqrt(d + e*x)) + A*c*(-16*d**(19/2)*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d
**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d**(19/2)/(3*d**8*e**3 +
9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 40*d**(17/2)*e*x*sqrt(1 +
e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 48
*d**(17/2)*e*x/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**
3) - 30*d**(15/2)*e**2*x**2*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**
6*e**5*x**2 + 3*d**5*e**6*x**3) + 48*d**(15/2)*e**2*x**2/(3*d**8*e**3 + 9*d**7*e
**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) - 4*d**(13/2)*e**3*x**3*sqrt(1 + e*
x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*x**3) + 16*d*
*(13/2)*e**3*x**3/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d**6*e**5*x**2 + 3*d**5*e**6*
x**3) + 2*d**(11/2)*e**4*x**4*sqrt(1 + e*x/d)/(3*d**8*e**3 + 9*d**7*e**4*x + 9*d
**6*e**5*x**2 + 3*d**5*e**6*x**3)) + B*a*Piecewise((4*d/(e**2*sqrt(d + e*x)) + 2
*x/(e*sqrt(d + e*x)), Ne(e, 0)), (x**2/(2*d**(3/2)), True)) + B*c*(32*d**(45/2)*
sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17
*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 32*
d**(45/2)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*
x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 176*d**(4
3/2)*e*x*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 +
100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x*
*6) - 192*d**(43/2)*e*x/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 1
00*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**
6) + 396*d**(41/2)*e**2*x**2*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 7
5*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**
5 + 5*d**14*e**10*x**6) - 480*d**(41/2)*e**2*x**2/(5*d**20*e**4 + 30*d**19*e**5*
x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**
9*x**5 + 5*d**14*e**10*x**6) + 462*d**(39/2)*e**3*x**3*sqrt(1 + e*x/d)/(5*d**20*
e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**
8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 640*d**(39/2)*e**3*x**3/(5*d
**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**1
6*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 290*d**(37/2)*e**4*x**4
*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**1
7*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 48
0*d**(37/2)*e**4*x**4/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100
*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6)
+ 92*d**(35/2)*e**5*x**5*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d
**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 +
5*d**14*e**10*x**6) - 192*d**(35/2)*e**5*x**5/(5*d**20*e**4 + 30*d**19*e**5*x +
75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x
**5 + 5*d**14*e**10*x**6) + 16*d**(33/2)*e**6*x**6*sqrt(1 + e*x/d)/(5*d**20*e**4
+ 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x*
*4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) - 32*d**(33/2)*e**6*x**6/(5*d**20*
e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*x**3 + 75*d**16*e**
8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 6*d**(31/2)*e**7*x**7*sqrt(1
+ e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**2 + 100*d**17*e**7*
x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**10*x**6) + 2*d**(29/
2)*e**8*x**8*sqrt(1 + e*x/d)/(5*d**20*e**4 + 30*d**19*e**5*x + 75*d**18*e**6*x**
2 + 100*d**17*e**7*x**3 + 75*d**16*e**8*x**4 + 30*d**15*e**9*x**5 + 5*d**14*e**1
0*x**6))
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GIAC/XCAS [A] time = 0.280138, size = 182, normalized size = 1.62 \[ \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} B c e^{16} - 15 \,{\left (x e + d\right )}^{\frac{3}{2}} B c d e^{16} + 45 \, \sqrt{x e + d} B c d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e^{17} - 30 \, \sqrt{x e + d} A c d e^{17} + 15 \, \sqrt{x e + d} B a e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (B c d^{3} - A c d^{2} e + B a d e^{2} - A a e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)*(B*x + A)/(e*x + d)^(3/2),x, algorithm="giac")
[Out]
2/15*(3*(x*e + d)^(5/2)*B*c*e^16 - 15*(x*e + d)^(3/2)*B*c*d*e^16 + 45*sqrt(x*e +
d)*B*c*d^2*e^16 + 5*(x*e + d)^(3/2)*A*c*e^17 - 30*sqrt(x*e + d)*A*c*d*e^17 + 15
*sqrt(x*e + d)*B*a*e^18)*e^(-20) + 2*(B*c*d^3 - A*c*d^2*e + B*a*d*e^2 - A*a*e^3)
*e^(-4)/sqrt(x*e + d)