3.106 \(\int \frac{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=1034 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{e+f x} (c+d x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^3 \sqrt{\frac{b (e+f x)}{b e-a f}} \sqrt{c+d x}}+\frac{2 \sqrt{d} (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{e+f x} \sqrt{c+d x}} \]

[Out]

(2*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e
+ B*c*f - 6*A*d*f) - a^2*b*(8*C*d*e + 6*c*C*f - B*d*f))*Sqrt[c + d*x]*Sqrt[e + f
*x])/(15*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(3/2)) - (2*(8*a^4*C*d^2*f^2 -
a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*
A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f -
3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f + 3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*
f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[c + d*x]*S
qrt[e + f*x])/(15*b^2*(b*c - a*d)^2*(b*e - a*f)^3*Sqrt[a + b*x]) - (2*(A*b^2 - a
*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b
*x)^(5/2)) + (2*Sqrt[d]*(8*a^4*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*
d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A
*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f
+ 3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*
C*e^2 - 13*f*(B*e - A*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt
icE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e
- a*f))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e +
f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e - c*f)*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d
*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(8*C*d*e +
6*c*C*f - B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)
]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/
(d*(b*e - a*f))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[
e + f*x])

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Rubi [A]  time = 9.58534, antiderivative size = 1034, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{e+f x} (c+d x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^3 \sqrt{\frac{b (e+f x)}{b e-a f}} \sqrt{c+d x}}+\frac{2 \sqrt{d} (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{e+f x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e
+ B*c*f - 6*A*d*f) - a^2*b*(8*C*d*e + 6*c*C*f - B*d*f))*Sqrt[c + d*x]*Sqrt[e + f
*x])/(15*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(3/2)) - (2*(8*a^4*C*d^2*f^2 -
a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*
A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f -
3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f + 3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*
f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[c + d*x]*S
qrt[e + f*x])/(15*b^2*(b*c - a*d)^2*(b*e - a*f)^3*Sqrt[a + b*x]) - (2*(A*b^2 - a
*(b*B - a*C))*(c + d*x)^(3/2)*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b
*x)^(5/2)) + (2*Sqrt[d]*(8*a^4*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*
d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A
*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f
+ 3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*
C*e^2 - 13*f*(B*e - A*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*Ellipt
icE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e
- a*f))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e +
f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e - c*f)*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d
*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(8*C*d*e +
6*c*C*f - B*d*f))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)
]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/
(d*(b*e - a*f))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[
e + f*x])

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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((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(7/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 20.9661, size = 13302, normalized size = 12.86 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

Result too large to show

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Maple [B]  time = 0.261, size = 33007, normalized size = 31.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((C*x^2 + B*x + A)*sqrt(d*x + c)/((b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a
^3)*sqrt(b*x + a)*sqrt(f*x + e)), 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((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(7/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), x)