3.764 \(\int \frac{x^4}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=250 \[ \frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2} d^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^3 d^3 (b c-a d)^2}+\frac{2 a x^3}{b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x} (a d+b c)}{b d \sqrt{c+d x} (b c-a d)^2} \]

[Out]

(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - (2*c*(b*c + a*d)*x^2*Sqr
t[a + b*x])/(b*d*(b*c - a*d)^2*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b
*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2) - 2*b*d*(5*b^2*c^2 - 2*a*b*c*d
+ 5*a^2*d^2)*x))/(4*b^3*d^3*(b*c - a*d)^2) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d
^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2)*d^(7/2)
)

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Rubi [A]  time = 0.626451, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{7/2} d^{7/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left ((a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )-2 b d x \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right )\right )}{4 b^3 d^3 (b c-a d)^2}+\frac{2 a x^3}{b \sqrt{a+b x} \sqrt{c+d x} (b c-a d)}-\frac{2 c x^2 \sqrt{a+b x} (a d+b c)}{b d \sqrt{c+d x} (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

(2*a*x^3)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - (2*c*(b*c + a*d)*x^2*Sqr
t[a + b*x])/(b*d*(b*c - a*d)^2*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c + d*x]*((b
*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 15*a^2*d^2) - 2*b*d*(5*b^2*c^2 - 2*a*b*c*d
+ 5*a^2*d^2)*x))/(4*b^3*d^3*(b*c - a*d)^2) + (3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d
^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(7/2)*d^(7/2)
)

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Rubi in Sympy [A]  time = 51.0912, size = 243, normalized size = 0.97 \[ - \frac{2 a x^{3}}{b \sqrt{a + b x} \sqrt{c + d x} \left (a d - b c\right )} - \frac{2 c x^{2} \sqrt{a + b x} \left (a d + b c\right )}{b d \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \sqrt{a + b x} \sqrt{c + d x} \left (- \frac{b d x \left (5 a^{2} d^{2} - 2 a b c d + 5 b^{2} c^{2}\right )}{4} + \left (\frac{a d}{8} + \frac{b c}{8}\right ) \left (15 a^{2} d^{2} - 22 a b c d + 15 b^{2} c^{2}\right )\right )}{b^{3} d^{3} \left (a d - b c\right )^{2}} + \frac{3 \left (5 a^{2} d^{2} + 6 a b c d + 5 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{7}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

-2*a*x**3/(b*sqrt(a + b*x)*sqrt(c + d*x)*(a*d - b*c)) - 2*c*x**2*sqrt(a + b*x)*(
a*d + b*c)/(b*d*sqrt(c + d*x)*(a*d - b*c)**2) - 2*sqrt(a + b*x)*sqrt(c + d*x)*(-
b*d*x*(5*a**2*d**2 - 2*a*b*c*d + 5*b**2*c**2)/4 + (a*d/8 + b*c/8)*(15*a**2*d**2
- 22*a*b*c*d + 15*b**2*c**2))/(b**3*d**3*(a*d - b*c)**2) + 3*(5*a**2*d**2 + 6*a*
b*c*d + 5*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(4*b**
(7/2)*d**(7/2))

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Mathematica [A]  time = 0.859054, size = 179, normalized size = 0.72 \[ \sqrt{a+b x} \sqrt{c+d x} \left (-\frac{2 a^4}{b^3 (a+b x) (b c-a d)^2}-\frac{7 (a d+b c)}{4 b^3 d^3}-\frac{2 c^4}{d^3 (c+d x) (a d-b c)^2}+\frac{x}{2 b^2 d^2}\right )+\frac{3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{7/2} d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/((a + b*x)^(3/2)*(c + d*x)^(3/2)),x]

[Out]

Sqrt[a + b*x]*Sqrt[c + d*x]*((-7*(b*c + a*d))/(4*b^3*d^3) + x/(2*b^2*d^2) - (2*a
^4)/(b^3*(b*c - a*d)^2*(a + b*x)) - (2*c^4)/(d^3*(-(b*c) + a*d)^2*(c + d*x))) +
(3*(5*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[
d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*b^(7/2)*d^(7/2))

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Maple [B]  time = 0.049, size = 1369, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^(3/2)/(d*x+c)^(3/2),x)

[Out]

1/8*(-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1
/2))*x^2*a^3*b^2*c*d^4+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+
a*d+b*c)/(b*d)^(1/2))*x*a^5*d^5+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*b^5*c^5+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^5*c*d^4+15*ln(1/2*(2*b*d*x+2*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a*b^4*c^5-30*x*a^4*d^4*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)-6*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^2*b^3*c^2*d^3-10*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)*x^2*a^3*b*d^4-10*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*b^4*c^3*d+14*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*a^3*b*c^2*d^2+14*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)*a^2*b^2*c^3*d-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b
*c)/(b*d)^(1/2))*x^2*a*b^4*c^3*d^2+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(
b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^4*b*c*d^4-18*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^3*b^2*c^2*d^3-18*ln(1/2*(2*b*d
*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*b^3*c^3*d^2
+3*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x
*a*b^4*c^4*d+4*x^3*a^2*b^2*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4*x^3*b^4*c^2
*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*x*b^4*c^4*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)-30*a^4*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-30*a*b^3*c^4*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(
1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*a^4*b*d^5+15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x^2*b^5*c^4*d-12*ln(1/2*(2*b*d*x+2*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*b*c^2*d^3-6*ln(1/2*(2*b
*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*b^2*c^3*d^2
-12*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*
a^2*b^3*c^4*d-8*x^3*a*b^3*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+10*((b*x+a)*
(d*x+c))^(1/2)*(b*d)^(1/2)*x^2*a^2*b^2*c*d^3+10*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1
/2)*x^2*a*b^3*c^2*d^2+4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^3*b*c*d^3+20*((b
*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*x*a^2*b^2*c^2*d^2+4*((b*x+a)*(d*x+c))^(1/2)*(b*
d)^(1/2)*x*a*b^3*c^3*d)/((b*x+a)*(d*x+c))^(1/2)/(b*d)^(1/2)/(a*d-b*c)^2/(b*x+a)^
(1/2)/(d*x+c)^(1/2)/b^3/d^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.658843, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="fricas")

[Out]

[-1/16*(4*(15*a*b^3*c^4 - 7*a^2*b^2*c^3*d - 7*a^3*b*c^2*d^2 + 15*a^4*c*d^3 - 2*(
b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^3 + 5*(b^4*c^3*d - a*b^3*c^2*d^2 -
a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + (15*b^4*c^4 - 2*a*b^3*c^3*d - 10*a^2*b^2*c^2*d^
2 - 2*a^3*b*c*d^3 + 15*a^4*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(5*
a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4
+ (5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b
*d^5)*x^2 + (5*b^5*c^5 + a*b^4*c^4*d - 6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a
^4*b*c*d^4 + 5*a^5*d^5)*x)*log(4*(2*b^2*d^2*x + b^2*c*d + a*b*d^2)*sqrt(b*x + a)
*sqrt(d*x + c) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a
*b*d^2)*x)*sqrt(b*d)))/((a*b^5*c^3*d^3 - 2*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + (b^
6*c^2*d^4 - 2*a*b^5*c*d^5 + a^2*b^4*d^6)*x^2 + (b^6*c^3*d^3 - a*b^5*c^2*d^4 - a^
2*b^4*c*d^5 + a^3*b^3*d^6)*x)*sqrt(b*d)), -1/8*(2*(15*a*b^3*c^4 - 7*a^2*b^2*c^3*
d - 7*a^3*b*c^2*d^2 + 15*a^4*c*d^3 - 2*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^
4)*x^3 + 5*(b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + (15*b^4
*c^4 - 2*a*b^3*c^3*d - 10*a^2*b^2*c^2*d^2 - 2*a^3*b*c*d^3 + 15*a^4*d^4)*x)*sqrt(
-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(5*a*b^4*c^5 - 4*a^2*b^3*c^4*d - 2*a^3*b^2
*c^3*d^2 - 4*a^4*b*c^2*d^3 + 5*a^5*c*d^4 + (5*b^5*c^4*d - 4*a*b^4*c^3*d^2 - 2*a^
2*b^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + 5*a^4*b*d^5)*x^2 + (5*b^5*c^5 + a*b^4*c^4*d -
6*a^2*b^3*c^3*d^2 - 6*a^3*b^2*c^2*d^3 + a^4*b*c*d^4 + 5*a^5*d^5)*x)*arctan(1/2*(
2*b*d*x + b*c + a*d)*sqrt(-b*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*d)))/((a*b^5*c^3*
d^3 - 2*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + (b^6*c^2*d^4 - 2*a*b^5*c*d^5 + a^2*b^4
*d^6)*x^2 + (b^6*c^3*d^3 - a*b^5*c^2*d^4 - a^2*b^4*c*d^5 + a^3*b^3*d^6)*x)*sqrt(
-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**(3/2)/(d*x+c)**(3/2),x)

[Out]

Integral(x**4/((a + b*x)**(3/2)*(c + d*x)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.590158, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((b*x + a)^(3/2)*(d*x + c)^(3/2)),x, algorithm="giac")

[Out]

sage0*x