3.1033 \(\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{x^{7/2}} \, dx\)

Optimal. Leaf size=421 \[ -\frac{2 \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 a b B-2 A \left (b^2-3 a c\right )\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 a^{7/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt [4]{c} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-3 \sqrt{a} A \sqrt{c}-5 a B+2 A b\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 a^{7/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{a+b x+c x^2} \left (-6 a A c-5 a b B+2 A b^2\right )}{15 a^2 \sqrt{x}}+\frac{2 \sqrt{c} \sqrt{x} \sqrt{a+b x+c x^2} \left (5 a b B-2 A \left (b^2-3 a c\right )\right )}{15 a^2 \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}} \]

[Out]

(2*(2*A*b^2 - 5*a*b*B - 6*a*A*c)*Sqrt[a + b*x + c*x^2])/(15*a^2*Sqrt[x]) - (2*(3
*a*A + (A*b + 5*a*B)*x)*Sqrt[a + b*x + c*x^2])/(15*a*x^(5/2)) + (2*Sqrt[c]*(5*a*
b*B - 2*A*(b^2 - 3*a*c))*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(15*a^2*(Sqrt[a] + Sqrt[
c]*x)) - (2*c^(1/4)*(5*a*b*B - 2*A*(b^2 - 3*a*c))*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a
+ b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(
1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*a^(7/4)*Sqrt[a + b*x + c*x^2]) - ((b +
2*Sqrt[a]*Sqrt[c])*(2*A*b - 5*a*B - 3*Sqrt[a]*A*Sqrt[c])*c^(1/4)*(Sqrt[a] + Sqrt
[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/
4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*a^(7/4)*Sqrt[a + b*x + c
*x^2])

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Rubi [A]  time = 1.00356, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{2 \sqrt [4]{c} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-6 a A c-5 a b B+2 A b^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 a^{7/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt [4]{c} \left (2 \sqrt{a} \sqrt{c}+b\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-3 \sqrt{a} A \sqrt{c}-5 a B+2 A b\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{15 a^{7/4} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{c} \sqrt{x} \sqrt{a+b x+c x^2} \left (-6 a A c-5 a b B+2 A b^2\right )}{15 a^2 \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{a+b x+c x^2} \left (5 a b B-2 A \left (b^2-3 a c\right )\right )}{15 a^2 \sqrt{x}}-\frac{2 \sqrt{a+b x+c x^2} (x (5 a B+A b)+3 a A)}{15 a x^{5/2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2*(5*a*b*B - 2*A*(b^2 - 3*a*c))*Sqrt[a + b*x + c*x^2])/(15*a^2*Sqrt[x]) - (2*(
3*a*A + (A*b + 5*a*B)*x)*Sqrt[a + b*x + c*x^2])/(15*a*x^(5/2)) - (2*Sqrt[c]*(2*A
*b^2 - 5*a*b*B - 6*a*A*c)*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(15*a^2*(Sqrt[a] + Sqrt
[c]*x)) + (2*c^(1/4)*(2*A*b^2 - 5*a*b*B - 6*a*A*c)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a
+ b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^
(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*a^(7/4)*Sqrt[a + b*x + c*x^2]) - ((b +
2*Sqrt[a]*Sqrt[c])*(2*A*b - 5*a*B - 3*Sqrt[a]*A*Sqrt[c])*c^(1/4)*(Sqrt[a] + Sqr
t[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1
/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(15*a^(7/4)*Sqrt[a + b*x +
c*x^2])

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Rubi in Sympy [A]  time = 144.988, size = 413, normalized size = 0.98 \[ - \frac{4 \left (\frac{3 A a}{2} + x \left (\frac{A b}{2} + \frac{5 B a}{2}\right )\right ) \sqrt{a + b x + c x^{2}}}{15 a x^{\frac{5}{2}}} - \frac{2 \sqrt{c} \sqrt{x} \sqrt{a + b x + c x^{2}} \left (- 6 A a c + 2 A b^{2} - 5 B a b\right )}{15 a^{2} \left (\sqrt{a} + \sqrt{c} x\right )} + \frac{4 \sqrt{a + b x + c x^{2}} \left (- 3 A a c + A b^{2} - \frac{5 B a b}{2}\right )}{15 a^{2} \sqrt{x}} + \frac{2 \sqrt [4]{c} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (- 6 A a c + 2 A b^{2} - 5 B a b\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{15 a^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} - \frac{\sqrt [4]{c} \sqrt{\frac{a + b x + c x^{2}}{\left (\sqrt{a} + \sqrt{c} x\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x\right ) \left (- 6 A a c + 2 A b^{2} - 5 B a b + \sqrt{a} \sqrt{c} \left (A b - 10 B a\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{15 a^{\frac{7}{4}} \sqrt{a + b x + c x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(7/2),x)

[Out]

-4*(3*A*a/2 + x*(A*b/2 + 5*B*a/2))*sqrt(a + b*x + c*x**2)/(15*a*x**(5/2)) - 2*sq
rt(c)*sqrt(x)*sqrt(a + b*x + c*x**2)*(-6*A*a*c + 2*A*b**2 - 5*B*a*b)/(15*a**2*(s
qrt(a) + sqrt(c)*x)) + 4*sqrt(a + b*x + c*x**2)*(-3*A*a*c + A*b**2 - 5*B*a*b/2)/
(15*a**2*sqrt(x)) + 2*c**(1/4)*sqrt((a + b*x + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)
*(sqrt(a) + sqrt(c)*x)*(-6*A*a*c + 2*A*b**2 - 5*B*a*b)*elliptic_e(2*atan(c**(1/4
)*sqrt(x)/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(15*a**(7/4)*sqrt(a + b*x + c*
x**2)) - c**(1/4)*sqrt((a + b*x + c*x**2)/(sqrt(a) + sqrt(c)*x)**2)*(sqrt(a) + s
qrt(c)*x)*(-6*A*a*c + 2*A*b**2 - 5*B*a*b + sqrt(a)*sqrt(c)*(A*b - 10*B*a))*ellip
tic_f(2*atan(c**(1/4)*sqrt(x)/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(15*a**(7/
4)*sqrt(a + b*x + c*x**2))

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Mathematica [C]  time = 7.47237, size = 576, normalized size = 1.37 \[ \frac{-4 (a+x (b+c x)) \left (a^2 (3 A+5 B x)+a x (A (b+6 c x)+5 b B x)-2 A b^2 x^2\right )+\frac{x^2 \left (i x^{3/2} \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{2 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+1} \sqrt{\frac{-2 x \sqrt{b^2-4 a c}+4 a+2 b x}{b x-x \sqrt{b^2-4 a c}}} \left (2 A \left (b^2-3 a c\right )-5 a b B\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+4 \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} (a+x (b+c x)) \left (6 a A c+5 a b B-2 A b^2\right )-i x^{3/2} \sqrt{\frac{2 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+1} \sqrt{\frac{-2 x \sqrt{b^2-4 a c}+4 a+2 b x}{b x-x \sqrt{b^2-4 a c}}} \left (2 A \left (b^2 \sqrt{b^2-4 a c}-3 a c \sqrt{b^2-4 a c}+4 a b c-b^3\right )+5 a B \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}}{30 a^2 x^{5/2} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*(a + x*(b + c*x))*(-2*A*b^2*x^2 + a^2*(3*A + 5*B*x) + a*x*(5*b*B*x + A*(b +
6*c*x))) + (x^2*(4*(-2*A*b^2 + 5*a*b*B + 6*a*A*c)*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])
]*(a + x*(b + c*x)) + I*(-b + Sqrt[b^2 - 4*a*c])*(-5*a*b*B + 2*A*(b^2 - 3*a*c))*
Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b
^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/
(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*
c])] - I*(5*a*B*(b^2 - 4*a*c - b*Sqrt[b^2 - 4*a*c]) + 2*A*(-b^3 + 4*a*b*c + b^2*
Sqrt[b^2 - 4*a*c] - 3*a*c*Sqrt[b^2 - 4*a*c]))*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*
a*c])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4
*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]]
, (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/Sqrt[a/(b + Sqrt[b^2 - 4*a*
c])])/(30*a^2*x^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.059, size = 2109, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(-12*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c
*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(
1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2
^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*x^2*a^2*c^2+24*A*((b+2
*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((
b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*
c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*x^2*a^2*c^2-5*B*((b+2*c*x+(-4*a*c+b^2)^
(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)
^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*
a*c+b^2)^(1/2))^(1/2))*x^2*a*b^3+2*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^
2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/
(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c
+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2
))*x^2*b^4-10*a^3*B*c*x-18*a^2*A*c^2*x^2-10*a^2*B*c^2*x^3-A*((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b
^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(
-4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a*b*c-6*A*((b+2*c*x+(-4*a*c+b^2
)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^
2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+
b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a*b*c-12*a*A*c^3*x^4-6*A*a^3*c+3
*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*
c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*Elli
pticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((
b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*x^2*a*b^2*c-14*A*((b+2*c*x+(-4*
a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4
*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(
-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1
/2))/(-4*a*c+b^2)^(1/2))^(1/2))*x^2*a*b^2*c+10*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(
b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^
(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2)
)/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)
^(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a^2*c-5*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b
+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(
1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))
/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^
(1/2))^(1/2))*(-4*a*c+b^2)^(1/2)*x^2*a*b^2+20*B*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b
+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(
1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))
/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^
(1/2))^(1/2))*x^2*a^2*b*c+2*A*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2
)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*
a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(
1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*(-4*
a*c+b^2)^(1/2)*x^2*b^3+4*A*x^4*b^2*c^2+4*A*x^3*b^3*c-10*B*x^4*a*b*c^2-14*A*x^3*a
*b*c^2-10*B*x^3*a*b^2*c+2*A*x^2*a*b^2*c-20*B*x^2*a^2*b*c-8*A*x*a^2*b*c)/(c*x^2+b
*x+a)^(1/2)/x^(5/2)/a^2/c

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2} + b x + a}{\left (B x + A\right )}}{x^{\frac{7}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**(7/2),x)

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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