3.1271 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=527 \[ -\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}-\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

[Out]

(-2*(d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*
c*Sqrt[b*x + c*x^2]) + (2*e*(30*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A
*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*b^2*c^3) +
(2*e*(10*A*c^2*d + 6*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(3/2)*Sqrt[b*x + c*
x^2])/(5*b^2*c^2) + (2*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e)
- 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B*d + 95*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)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*d*(c*d -
b*e)*(30*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d +
20*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)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[d + e*x]*Sqrt[b*x
+ c*x^2])

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Rubi [A]  time = 2.18308, antiderivative size = 527, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 (d+e x)^{5/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \sqrt{b x+c x^2}}+\frac{2 e \sqrt{b x+c x^2} (d+e x)^{3/2} \left (-5 b c (A e+B d)+10 A c^2 d+6 b^2 B e\right )}{5 b^2 c^2}+\frac{2 e \sqrt{b x+c x^2} \sqrt{d+e x} \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right )}{15 b^2 c^3}-\frac{2 d \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} (c d-b e) \left (b^2 c e (20 A e+43 B d)-15 b c^2 d (2 A e+B d)+30 A c^3 d^2-24 b^3 B e^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 (-b)^{3/2} c^{7/2} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(-2*(d + e*x)^(5/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(b^2*
c*Sqrt[b*x + c*x^2]) + (2*e*(30*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A
*e) + b^2*c*e*(43*B*d + 20*A*e))*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(15*b^2*c^3) +
(2*e*(10*A*c^2*d + 6*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(3/2)*Sqrt[b*x + c*
x^2])/(5*b^2*c^2) + (2*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e)
- 8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B*d + 95*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)])/(15*(-b)^(3/2)*c^(7/2)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*d*(c*d -
b*e)*(30*A*c^3*d^2 - 24*b^3*B*e^2 - 15*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(43*B*d +
20*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)])/(15*(-b)^(3/2)*c^(7/2)*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)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 7.30363, size = 493, normalized size = 0.94 \[ \frac{2 \left (b (d+e x) \left (b^2 e^2 x (b+c x) (5 A c e-9 b B e+16 B c d)+15 x (b B-A c) (c d-b e)^3-15 A c^3 d^3 (b+c x)+3 b^2 B c e^3 x^2 (b+c x)\right )+\sqrt{\frac{b}{c}} \left (-i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) \left (8 b^2 c e (5 A e+13 B d)-15 b c^2 d (5 A e+4 B d)+15 A c^3 d^2-48 b^3 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (-8 b^3 c e^2 (5 A e+16 B d)+b^2 c^2 d e (95 A e+103 B d)-15 b c^3 d^2 (3 A e+B d)+30 A c^4 d^3+48 b^4 B e^3\right )\right )\right )}{15 b^3 c^3 \sqrt{x (b+c x)} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(7/2))/(b*x + c*x^2)^(3/2),x]

[Out]

(2*(b*(d + e*x)*(15*(b*B - A*c)*(c*d - b*e)^3*x - 15*A*c^3*d^3*(b + c*x) + b^2*e
^2*(16*B*c*d - 9*b*B*e + 5*A*c*e)*x*(b + c*x) + 3*b^2*B*c*e^3*x^2*(b + c*x)) + S
qrt[b/c]*(Sqrt[b/c]*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e) -
8*b^3*c*e^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B*d + 95*A*e))*(b + c*x)*(d + e*
x) + I*b*e*(30*A*c^4*d^3 + 48*b^4*B*e^3 - 15*b*c^3*d^2*(B*d + 3*A*e) - 8*b^3*c*e
^2*(16*B*d + 5*A*e) + b^2*c^2*d*e*(103*B*d + 95*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*(
c*d - b*e)*(15*A*c^3*d^2 - 48*b^3*B*e^2 - 15*b*c^2*d*(4*B*d + 5*A*e) + 8*b^2*c*e
*(13*B*d + 5*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSi
nh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*b^3*c^3*Sqrt[x*(b + c*x)]*Sqrt[d + e*
x])

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Maple [B]  time = 0.055, size = 1766, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(7/2)/(c*x^2+b*x)^(3/2),x)

[Out]

2/15*(45*A*x^2*b*c^5*d^2*e^2-30*A*x*c^6*d^4-30*A*x^2*c^6*d^3*e-6*B*x^3*b^3*c^3*e
^4-24*B*x^2*b^4*c^2*e^4+5*A*x^3*b^2*c^4*e^4+20*A*x^2*b^3*c^3*e^4+3*B*x^4*b^2*c^4
*e^4-15*A*b*c^5*d^4+15*B*x*b*c^5*d^4+40*e^4*b^5*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
))*c+30*b*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*Ellipt
icE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*c^5*d^4+15*b^2*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))*c^4*d^4-15*b^2*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))*c^4*d^4-48*e^4*
b^6*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))+15*B*x^2*b*c^5*d^3*e-45*A*x*b^2*c^4*d^2*e
^2-45*B*x*b^2*c^4*d^3*e+20*A*x*b^3*c^3*d*e^3+55*B*x^2*b^3*c^3*d*e^3-24*B*x*b^4*c
^2*d*e^3-40*A*x^2*b^2*c^4*d*e^3+30*A*x*b*c^5*d^3*e+61*B*x*b^3*c^3*d^2*e^2+19*B*x
^3*b^2*c^4*d*e^3-29*B*x^2*b^2*c^4*d^2*e^2-30*b*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)
)*c^5*d^4+20*e^3*b^4*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))*c^2*d-50*e^2*b^3*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))*c^3*d^2+60*e*b^2*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))*
c^4*d^3-135*e^3*b^4*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))*c^2*d+140*e^2*b^3*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))*c^3*d^2-75*e*b^2*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))*
c^4*d^3-24*e^3*b^5*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))*c*d+67*e^2*b^4*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))*c^2*d^2-58*e*b^3*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))*c^3*
d^3+176*e^3*b^5*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))*c*d-231*e^2*b^4*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))*c^2*d^2+118*e*b^3*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))*c^3*d
^3)/x*(x*(c*x+b))^(1/2)/(c*x+b)/c^5/b^2/(e*x+d)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B e^{3} x^{4} + A d^{3} +{\left (3 \, B d e^{2} + A e^{3}\right )} x^{3} + 3 \,{\left (B d^{2} e + A d e^{2}\right )} x^{2} +{\left (B d^{3} + 3 \, A d^{2} e\right )} x\right )} \sqrt{e x + d}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")

[Out]

integral((B*e^3*x^4 + A*d^3 + (3*B*d*e^2 + A*e^3)*x^3 + 3*(B*d^2*e + A*d*e^2)*x^
2 + (B*d^3 + 3*A*d^2*e)*x)*sqrt(e*x + d)/(c*x^2 + b*x)^(3/2), 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)**(7/2)/(c*x**2+b*x)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{7}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")

[Out]

integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 + b*x)^(3/2), x)