3.1264 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\sqrt{b x+c x^2}} \, dx\)

Optimal. Leaf size=460 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (28 A c e (2 c d-b e)+B \left (24 b^2 e^2-43 b c d e+15 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (28 A c e (2 c d-b e)+B \left (24 b^2 e^2-43 b c d e+15 c^2 d^2\right )\right )}{105 c^3}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (7 A c e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B \left (-48 b^3 e^3+128 b^2 c d e^2-103 b c^2 d^2 e+15 c^3 d^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (d+e x)^{3/2} (7 A c e-6 b B e+5 B c d)}{35 c^2}+\frac{2 B \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]

[Out]

(2*(28*A*c*e*(2*c*d - b*e) + B*(15*c^2*d^2 - 43*b*c*d*e + 24*b^2*e^2))*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])/(105*c^3) + (2*(5*B*c*d - 6*b*B*e + 7*A*c*e)*(d + e*x)^(
3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*B*(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c)
+ (2*Sqrt[-b]*(7*A*c*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2) + B*(15*c^3*d^3 -
103*b*c^2*d^2*e + 128*b^2*c*d*e^2 - 48*b^3*e^3))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[
d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(7/2
)*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(28*A*c*e*(
2*c*d - b*e) + B*(15*c^2*d^2 - 43*b*c*d*e + 24*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/
b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])
/(105*c^(7/2)*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.84288, antiderivative size = 460, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \sqrt{-b} d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (28 A c e (2 c d-b e)+B \left (24 b^2 e^2-43 b c d e+15 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} e \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{b x+c x^2} \sqrt{d+e x} \left (28 A c e (2 c d-b e)+B \left (24 b^2 e^2-43 b c d e+15 c^2 d^2\right )\right )}{105 c^3}+\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (7 A c e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B \left (-48 b^3 e^3+128 b^2 c d e^2-103 b c^2 d^2 e+15 c^3 d^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{105 c^{7/2} e \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}+\frac{2 \sqrt{b x+c x^2} (d+e x)^{3/2} (7 A c e-6 b B e+5 B c d)}{35 c^2}+\frac{2 B \sqrt{b x+c x^2} (d+e x)^{5/2}}{7 c} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(5/2))/Sqrt[b*x + c*x^2],x]

[Out]

(2*(28*A*c*e*(2*c*d - b*e) + B*(15*c^2*d^2 - 43*b*c*d*e + 24*b^2*e^2))*Sqrt[d +
e*x]*Sqrt[b*x + c*x^2])/(105*c^3) + (2*(5*B*c*d - 6*b*B*e + 7*A*c*e)*(d + e*x)^(
3/2)*Sqrt[b*x + c*x^2])/(35*c^2) + (2*B*(d + e*x)^(5/2)*Sqrt[b*x + c*x^2])/(7*c)
+ (2*Sqrt[-b]*(7*A*c*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2) + B*(15*c^3*d^3 -
103*b*c^2*d^2*e + 128*b^2*c*d*e^2 - 48*b^3*e^3))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[
d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(105*c^(7/2
)*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*(28*A*c*e*(
2*c*d - b*e) + B*(15*c^2*d^2 - 43*b*c*d*e + 24*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/
b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])
/(105*c^(7/2)*e*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)**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.20626, size = 479, normalized size = 1.04 \[ \frac{2 \sqrt{x} \left (\sqrt{x} (b+c x) (d+e x) \left (7 A c e (-4 b e+11 c d+3 c e x)+B \left (24 b^2 e^2-b c e (61 d+18 e x)+15 c^2 \left (3 d^2+3 d e x+e^2 x^2\right )\right )\right )+\frac{i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (b e-c d) \left (-8 b^2 c e (7 A e+13 B d)+b c^2 d (133 A e+60 B d)-105 A c^3 d^2+48 b^3 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )}{b}+\frac{(b+c x) (d+e x) \left (7 A c e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B \left (-48 b^3 e^3+128 b^2 c d e^2-103 b c^2 d^2 e+15 c^3 d^3\right )\right )}{c e \sqrt{x}}+i x \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (7 A c e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )+B \left (-48 b^3 e^3+128 b^2 c d e^2-103 b c^2 d^2 e+15 c^3 d^3\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )}{105 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/Sqrt[b*x + c*x^2],x]

[Out]

(2*Sqrt[x]*(((7*A*c*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2) + B*(15*c^3*d^3 - 10
3*b*c^2*d^2*e + 128*b^2*c*d*e^2 - 48*b^3*e^3))*(b + c*x)*(d + e*x))/(c*e*Sqrt[x]
) + Sqrt[x]*(b + c*x)*(d + e*x)*(7*A*c*e*(11*c*d - 4*b*e + 3*c*e*x) + B*(24*b^2*
e^2 - b*c*e*(61*d + 18*e*x) + 15*c^2*(3*d^2 + 3*d*e*x + e^2*x^2))) + I*Sqrt[b/c]
*(7*A*c*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2) + B*(15*c^3*d^3 - 103*b*c^2*d^2*
e + 128*b^2*c*d*e^2 - 48*b^3*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*Ellipti
cE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (I*Sqrt[b/c]*(-(c*d) + b*e)*(-10
5*A*c^3*d^2 + 48*b^3*B*e^2 - 8*b^2*c*e*(13*B*d + 7*A*e) + b*c^2*d*(60*B*d + 133*
A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x
]], (c*d)/(b*e)])/b))/(105*c^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])

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Maple [B]  time = 0.039, size = 1610, normalized size = 3.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(5/2)/(c*x^2+b*x)^(1/2),x)

[Out]

-2/105*(e*x+d)^(1/2)*(x*(c*x+b))^(1/2)*(-24*B*((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^3+67*B*((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^3*c^2*d^2*e^2-58*B*((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^3*e+176*B*((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+118*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^3*e-15*B*((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^4+56*A*((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^4+19*
B*x^3*b*c^4*d*e^3+3*B*x^4*b*c^4*e^4-6*B*x^3*b^2*c^3*e^4-90*B*x^3*c^5*d^2*e^2+28*
A*x^2*b^2*c^3*e^4-77*A*x^2*c^5*d^2*e^2-24*B*x^2*b^3*c^2*e^4-45*B*x^2*c^5*d^3*e-7
0*A*x^2*b*c^4*d*e^3+55*B*x^2*b^2*c^3*d*e^3-29*B*x^2*b*c^4*d^2*e^2+28*A*x*b^2*c^3
*d*e^3-77*A*x*b*c^4*d^2*e^2-24*B*x*b^3*c^2*d*e^3+61*B*x*b^2*c^3*d^2*e^2-45*B*x*b
*c^4*d^3*e-48*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*El
lipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^5*e^4+15*B*((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^4-60*B*x^4*c^5*d*e^3+7*A*x^3*b*c^4*e^4-98*A*x^3*c^5*d*e^3
-21*A*x^4*c^5*e^4-15*B*x^5*c^5*e^4+28*A*((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^3-84*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2*e^2+56*A*((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^3*e-217*A*((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^3+322*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Elliptic
E(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^3*d^2*e^2-161*A*((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^3*e-231*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*b^3*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2))/e/c^5/x/(c*e*x^2+b*e*x+c*d*x+b*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(c*x^2 + b*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{2} x^{3} + A d^{2} +{\left (2 \, B d e + A e^{2}\right )} x^{2} +{\left (B d^{2} + 2 \, A d e\right )} x\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")

[Out]

integral((B*e^2*x^3 + A*d^2 + (2*B*d*e + A*e^2)*x^2 + (B*d^2 + 2*A*d*e)*x)*sqrt(
e*x + d)/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)**(5/2)/(c*x**2+b*x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{\sqrt{c x^{2} + b x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(5/2)/sqrt(c*x^2 + b*x), x)