3.1280 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx\)
Optimal. Leaf size=410 \[ \frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (3 A e+5 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{d+e x} \left (c x \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right )+b \left (-b c (5 A e+4 B d)+8 A c^2 d+b^2 B e\right )\right )}{3 b^4 c \sqrt{b x+c x^2}} \]
[Out]
(-2*Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*
c*(b*x + c*x^2)^(3/2)) + (2*Sqrt[d + e*x]*(b*(8*A*c^2*d + b^2*B*e - b*c*(4*B*d +
5*A*e)) + c*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d + A*e))*x))/(3*b^4*c*Sqrt[b*x +
c*x^2]) - (2*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b
]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(
-b)^(7/2)*Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(16*A*c^2*d^2 - 8*b*
c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 3*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e
*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)
*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi [A] time = 1.47316, antiderivative size = 410, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286
\[ \frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (b^2 e (3 A e+5 B d)-8 b c d (2 A e+B d)+16 A c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{3 b^2 c \left (b x+c x^2\right )^{3/2}}-\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{3 (-b)^{7/2} \sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{d+e x} \left (b \left (5 A b c e-8 A c^2 d+b^2 (-B) e+4 b B c d\right )-c x \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right )\right )}{3 b^4 c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*Sqrt[d + e*x]*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(3*b^2*
c*(b*x + c*x^2)^(3/2)) - (2*Sqrt[d + e*x]*(b*(4*b*B*c*d - 8*A*c^2*d - b^2*B*e +
5*A*b*c*e) - c*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d + A*e))*x))/(3*b^4*c*Sqrt[b*x
+ c*x^2]) - (2*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d + A*e))*Sqrt[x]*Sqrt[1 + (c*x)
/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3
*(-b)^(7/2)*Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*(16*A*c^2*d^2 - 8*
b*c*d*(B*d + 2*A*e) + b^2*e*(5*B*d + 3*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 +
(e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/
2)*Sqrt[c]*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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Mathematica [C] time = 3.2394, size = 378, normalized size = 0.92 \[ -\frac{2 \left (b (d+e x) \left (A \left (b^3 (d+4 e x)+b^2 c x (13 e x-6 d)+8 b c^2 x^2 (e x-3 d)-16 c^3 d x^3\right )+b B x \left (b^2 (3 d-2 e x)+b c x (12 d-e x)+8 c^2 d x^2\right )\right )+x \sqrt{\frac{b}{c}} (b+c x) \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-b c (5 A e+4 B d)+8 A c^2 d+b^2 B e\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b c (A e+B d)+16 A c^2 d+b^2 B e\right )\right )\right )}{3 b^5 (x (b+c x))^{3/2} \sqrt{d+e x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x)^(3/2))/(b*x + c*x^2)^(5/2),x]
[Out]
(-2*(b*(d + e*x)*(b*B*x*(8*c^2*d*x^2 + b^2*(3*d - 2*e*x) + b*c*x*(12*d - e*x)) +
A*(-16*c^3*d*x^3 + 8*b*c^2*x^2*(-3*d + e*x) + b^3*(d + 4*e*x) + b^2*c*x*(-6*d +
13*e*x))) + Sqrt[b/c]*x*(b + c*x)*(Sqrt[b/c]*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d
+ A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^2*d + b^2*B*e - 8*b*c*(B*d + A*e))*
Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x
]], (c*d)/(b*e)] - I*b*e*(8*A*c^2*d + b^2*B*e - b*c*(4*B*d + 5*A*e))*Sqrt[1 + b/
(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(
b*e)])))/(3*b^5*(x*(b + c*x))^(3/2)*Sqrt[d + e*x])
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Maple [B] time = 0.06, size = 2043, normalized size = 5. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x)
[Out]
2/3*(-11*B*x^3*b^2*c^3*d*e-8*B*x^3*b*c^4*d^2+16*A*x^3*c^5*d^2-A*b^3*c^2*d^2+24*A
*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d*e+5*B*x^2*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*b^3*c^2*d*e-9*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e-16*A
*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d*e+24*A*x*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))
^(1/2))*b^3*c^2*d*e+5*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e-9*B*x*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(
1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*d*e+3*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b
^3*c^2*e^2+16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^4*d^2-8*A*x^2*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*b^3*c^2*e^2-16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-
c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*
c^4*d^2-8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2+B*x^2*((c*x+b)/b)^(
1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^4*c*e^2+8*B*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(
1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d
^2+3*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*e^2+16*A*x*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*b^2*c^3*d^2-8*A*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*c*e^2-16*A*x*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2-8*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*b^3*c^2*d^2+8*B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1
/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c^2*d^2+24*A*x^2*b*c^
4*d^2-3*B*x*b^3*c^2*d^2-8*A*x^4*b*c^4*e^2+16*A*x^4*c^5*d*e-13*A*x^3*b^2*c^3*e^2-
12*B*x^2*b^2*c^3*d^2+6*A*x*b^2*c^3*d^2+2*B*x^3*b^3*c^2*e^2+B*x^4*b^2*c^3*e^2-4*A
*x^2*b^3*c^2*e^2-7*A*x^2*b^2*c^3*d*e-B*x^2*b^3*c^2*d*e-5*A*x*b^3*c^2*d*e-8*B*x^4
*b*c^4*d*e+16*A*x^3*b*c^4*d*e-16*A*x^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^
(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*
d*e+B*x*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^2)/x^2*(x*(c*x+b))^(1/2)/b^4/c^2/
(c*x+b)^2/(e*x+d)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e x^{2} + A d +{\left (B d + A e\right )} x\right )} \sqrt{e x + d}}{{\left (c^{2} x^{4} + 2 \, b c x^{3} + b^{2} x^{2}\right )} \sqrt{c x^{2} + b x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
integral((B*e*x^2 + A*d + (B*d + A*e)*x)*sqrt(e*x + d)/((c^2*x^4 + 2*b*c*x^3 + b
^2*x^2)*sqrt(c*x^2 + b*x)), x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)
[Out]
Timed out
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]
integrate((B*x + A)*(e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)