3.736 \(\int \frac{x^4}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=258 \[ \frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{12 b^2 d^4 (b c-a d)^2}-\frac{2 c x^2 \sqrt{a+b x} (7 b c-9 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{2 c x^3 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

(-2*c*x^3*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - (2*c*(7*b*c - 9*a*d
)*x^2*Sqrt[a + b*x])/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c
+ d*x]*(105*b^3*c^3 - 145*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 9*a^3*d^3 - 2*b*d*(35*
b^2*c^2 - 46*a*b*c*d + 3*a^2*d^2)*x))/(12*b^2*d^4*(b*c - a*d)^2) + ((35*b^2*c^2
+ 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x]
)])/(4*b^(5/2)*d^(9/2))

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Rubi [A]  time = 0.620708, antiderivative size = 258, 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 \[ \frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{5/2} d^{9/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{12 b^2 d^4 (b c-a d)^2}-\frac{2 c x^2 \sqrt{a+b x} (7 b c-9 a d)}{3 d^2 \sqrt{c+d x} (b c-a d)^2}-\frac{2 c x^3 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(-2*c*x^3*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) - (2*c*(7*b*c - 9*a*d
)*x^2*Sqrt[a + b*x])/(3*d^2*(b*c - a*d)^2*Sqrt[c + d*x]) - (Sqrt[a + b*x]*Sqrt[c
+ d*x]*(105*b^3*c^3 - 145*a*b^2*c^2*d + 15*a^2*b*c*d^2 + 9*a^3*d^3 - 2*b*d*(35*
b^2*c^2 - 46*a*b*c*d + 3*a^2*d^2)*x))/(12*b^2*d^4*(b*c - a*d)^2) + ((35*b^2*c^2
+ 10*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x]
)])/(4*b^(5/2)*d^(9/2))

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Rubi in Sympy [A]  time = 53.4482, size = 262, normalized size = 1.02 \[ \frac{2 c x^{3} \sqrt{a + b x}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{2 c x^{2} \sqrt{a + b x} \left (9 a d - 7 b c\right )}{3 d^{2} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{2 \sqrt{a + b x} \sqrt{c + d x} \left (\frac{9 a^{3} d^{3}}{8} + \frac{15 a^{2} b c d^{2}}{8} - \frac{145 a b^{2} c^{2} d}{8} + \frac{105 b^{3} c^{3}}{8} - \frac{b d x \left (3 a^{2} d^{2} - 46 a b c d + 35 b^{2} c^{2}\right )}{4}\right )}{3 b^{2} d^{4} \left (a d - b c\right )^{2}} + \frac{\left (3 a^{2} d^{2} + 10 a b c d + 35 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{5}{2}} d^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

2*c*x**3*sqrt(a + b*x)/(3*d*(c + d*x)**(3/2)*(a*d - b*c)) + 2*c*x**2*sqrt(a + b*
x)*(9*a*d - 7*b*c)/(3*d**2*sqrt(c + d*x)*(a*d - b*c)**2) - 2*sqrt(a + b*x)*sqrt(
c + d*x)*(9*a**3*d**3/8 + 15*a**2*b*c*d**2/8 - 145*a*b**2*c**2*d/8 + 105*b**3*c*
*3/8 - b*d*x*(3*a**2*d**2 - 46*a*b*c*d + 35*b**2*c**2)/4)/(3*b**2*d**4*(a*d - b*
c)**2) + (3*a**2*d**2 + 10*a*b*c*d + 35*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(
sqrt(b)*sqrt(c + d*x)))/(4*b**(5/2)*d**(9/2))

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Mathematica [A]  time = 0.464395, size = 181, normalized size = 0.7 \[ \frac{\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{5/2} d^{9/2}}+\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-\frac{9 a d+33 b c}{b^2}+\frac{8 c^4}{(c+d x)^2 (b c-a d)}-\frac{16 c^3 (5 b c-6 a d)}{(c+d x) (b c-a d)^2}+\frac{6 d x}{b}\right )}{12 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(-((33*b*c + 9*a*d)/b^2) + (6*d*x)/b + (8*c^4)/((b*
c - a*d)*(c + d*x)^2) - (16*c^3*(5*b*c - 6*a*d))/((b*c - a*d)^2*(c + d*x))))/(12
*d^4) + ((35*b^2*c^2 + 10*a*b*c*d + 3*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^(5/2)*d^(9/2))

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Maple [B]  time = 0.048, size = 1287, normalized size = 5. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(d*x+c)^(5/2)/(b*x+a)^(1/2),x)

[Out]

1/24*(b*x+a)^(1/2)*(-18*x^2*a^3*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-18*a^3*c
^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+105*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^4*c^4*d^2+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^4*c*d^5+210*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^4*c^5*
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^3*b*c^3*d^3+54*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^2*c^4*d^2-210*b^3*c^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+9*
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*d^6+9*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*c^2*d^4-180*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^3*c^5*d+105*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))*b^4*c^6+108*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^2*c^3*d^3-360*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^3*c^4*d^2+12*x^3*a
^2*b*d^5*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+12*x^3*b^3*c^2*d^3*((b*x+a)*(d*x+c)
)^(1/2)*(b*d)^(1/2)-42*x^2*b^3*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-36*x*
a^3*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-280*x*b^3*c^4*d*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)-30*a^2*b*c^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+290*a*b^2
*c^4*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*x^2*a^3*b*c*d^5+54*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^2*c^2*d^4-180
*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^3*c^3*d^3+24*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*c^2*d^4+66*x^2*a*b^2*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)
^(1/2)-48*x*a^2*b*c^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+396*x*a*b^2*c^3*d^
2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-24*x^3*a*b^2*c*d^4*((b*x+a)*(d*x+c))^(1/2)
*(b*d)^(1/2)-6*x^2*a^2*b*c*d^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(a*d-b*c)^2/
(b*d)^(1/2)/b^2/((b*x+a)*(d*x+c))^(1/2)/d^4/(d*x+c)^(3/2)

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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/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.7431, 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/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[-1/48*(4*(105*b^3*c^5 - 145*a*b^2*c^4*d + 15*a^2*b*c^3*d^2 + 9*a^3*c^2*d^3 - 6*
(b^3*c^2*d^3 - 2*a*b^2*c*d^4 + a^2*b*d^5)*x^3 + 3*(7*b^3*c^3*d^2 - 11*a*b^2*c^2*
d^3 + a^2*b*c*d^4 + 3*a^3*d^5)*x^2 + 2*(70*b^3*c^4*d - 99*a*b^2*c^3*d^2 + 12*a^2
*b*c^2*d^3 + 9*a^3*c*d^4)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(35*b^4*c
^6 - 60*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 + 4*a^3*b*c^3*d^3 + 3*a^4*c^2*d^4 + (35
*b^4*c^4*d^2 - 60*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d^4 + 4*a^3*b*c*d^5 + 3*a^4*d^6
)*x^2 + 2*(35*b^4*c^5*d - 60*a*b^3*c^4*d^2 + 18*a^2*b^2*c^3*d^3 + 4*a^3*b*c^2*d^
4 + 3*a^4*c*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)))/((b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 + a^2*b^2*c^2*d^6 + (b^4*c^2*d^6
- 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*(b^4*c^3*d^5 - 2*a*b^3*c^2*d^6 + a^2*b^2
*c*d^7)*x)*sqrt(b*d)), -1/24*(2*(105*b^3*c^5 - 145*a*b^2*c^4*d + 15*a^2*b*c^3*d^
2 + 9*a^3*c^2*d^3 - 6*(b^3*c^2*d^3 - 2*a*b^2*c*d^4 + a^2*b*d^5)*x^3 + 3*(7*b^3*c
^3*d^2 - 11*a*b^2*c^2*d^3 + a^2*b*c*d^4 + 3*a^3*d^5)*x^2 + 2*(70*b^3*c^4*d - 99*
a*b^2*c^3*d^2 + 12*a^2*b*c^2*d^3 + 9*a^3*c*d^4)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt
(d*x + c) - 3*(35*b^4*c^6 - 60*a*b^3*c^5*d + 18*a^2*b^2*c^4*d^2 + 4*a^3*b*c^3*d^
3 + 3*a^4*c^2*d^4 + (35*b^4*c^4*d^2 - 60*a*b^3*c^3*d^3 + 18*a^2*b^2*c^2*d^4 + 4*
a^3*b*c*d^5 + 3*a^4*d^6)*x^2 + 2*(35*b^4*c^5*d - 60*a*b^3*c^4*d^2 + 18*a^2*b^2*c
^3*d^3 + 4*a^3*b*c^2*d^4 + 3*a^4*c*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)))/((b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 + a^2
*b^2*c^2*d^6 + (b^4*c^2*d^6 - 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*(b^4*c^3*d^5
- 2*a*b^3*c^2*d^6 + a^2*b^2*c*d^7)*x)*sqrt(-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral(x**4/(sqrt(a + b*x)*(c + d*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.284734, size = 686, normalized size = 2.66 \[ \frac{{\left ({\left (3 \,{\left (b x + a\right )}{\left (\frac{2 \,{\left (b^{7} c^{2} d^{6} - 2 \, a b^{6} c d^{7} + a^{2} b^{5} d^{8}\right )}{\left (b x + a\right )}}{b^{7} c^{2} d^{7}{\left | b \right |} - 2 \, a b^{6} c d^{8}{\left | b \right |} + a^{2} b^{5} d^{9}{\left | b \right |}} - \frac{7 \, b^{8} c^{3} d^{5} - 5 \, a b^{7} c^{2} d^{6} - 11 \, a^{2} b^{6} c d^{7} + 9 \, a^{3} b^{5} d^{8}}{b^{7} c^{2} d^{7}{\left | b \right |} - 2 \, a b^{6} c d^{8}{\left | b \right |} + a^{2} b^{5} d^{9}{\left | b \right |}}\right )} - \frac{4 \,{\left (35 \, b^{9} c^{4} d^{4} - 60 \, a b^{8} c^{3} d^{5} + 18 \, a^{2} b^{7} c^{2} d^{6} + 12 \, a^{3} b^{6} c d^{7} - 9 \, a^{4} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7}{\left | b \right |} - 2 \, a b^{6} c d^{8}{\left | b \right |} + a^{2} b^{5} d^{9}{\left | b \right |}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (35 \, b^{10} c^{5} d^{3} - 95 \, a b^{9} c^{4} d^{4} + 78 \, a^{2} b^{8} c^{3} d^{5} - 14 \, a^{3} b^{7} c^{2} d^{6} - 9 \, a^{4} b^{6} c d^{7} + 5 \, a^{5} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7}{\left | b \right |} - 2 \, a b^{6} c d^{8}{\left | b \right |} + a^{2} b^{5} d^{9}{\left | b \right |}}\right )} \sqrt{b x + a}}{12 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} - \frac{{\left (35 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt{b d} b d^{4}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

1/12*((3*(b*x + a)*(2*(b^7*c^2*d^6 - 2*a*b^6*c*d^7 + a^2*b^5*d^8)*(b*x + a)/(b^7
*c^2*d^7*abs(b) - 2*a*b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)) - (7*b^8*c^3*d^5 -
5*a*b^7*c^2*d^6 - 11*a^2*b^6*c*d^7 + 9*a^3*b^5*d^8)/(b^7*c^2*d^7*abs(b) - 2*a*b^
6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b))) - 4*(35*b^9*c^4*d^4 - 60*a*b^8*c^3*d^5 + 1
8*a^2*b^7*c^2*d^6 + 12*a^3*b^6*c*d^7 - 9*a^4*b^5*d^8)/(b^7*c^2*d^7*abs(b) - 2*a*
b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)))*(b*x + a) - 3*(35*b^10*c^5*d^3 - 95*a*b^
9*c^4*d^4 + 78*a^2*b^8*c^3*d^5 - 14*a^3*b^7*c^2*d^6 - 9*a^4*b^6*c*d^7 + 5*a^5*b^
5*d^8)/(b^7*c^2*d^7*abs(b) - 2*a*b^6*c*d^8*abs(b) + a^2*b^5*d^9*abs(b)))*sqrt(b*
x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) - 1/4*(35*b^2*c^2 + 10*a*b*c*d + 3*
a^2*d^2)*ln(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + a)*b*d - a*b*d)))
/(sqrt(b*d)*b*d^4*abs(b))