3.737 \(\int \frac{x^3}{\sqrt{a+b x} (c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=174 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt{c+d x} (b c-a d)^2}-\frac{(a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac{2 c x^2 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.373151, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt{c+d x} (b c-a d)^2}-\frac{(a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac{2 c x^2 \sqrt{a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 27.2585, size = 165, normalized size = 0.95 \[ \frac{2 c x^{2} \sqrt{a + b x}}{3 d \left (c + d x\right )^{\frac{3}{2}} \left (a d - b c\right )} + \frac{4 \sqrt{a + b x} \left (\frac{c \left (3 a^{2} d^{2} - 22 a b c d + 15 b^{2} c^{2}\right )}{4} + \frac{d x \left (a d - b c\right ) \left (3 a d - 5 b c\right )}{4}\right )}{3 b d^{3} \sqrt{c + d x} \left (a d - b c\right )^{2}} - \frac{\left (a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{b^{\frac{3}{2}} d^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.546933, size = 150, normalized size = 0.86 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (\frac{2 c^3}{(c+d x)^2 (a d-b c)}+\frac{2 c^2 (7 b c-9 a d)}{(c+d x) (b c-a d)^2}+\frac{3}{b}\right )}{3 d^3}-\frac{(a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{2 b^{3/2} d^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(Sqrt[a + b*x]*(c + d*x)^(5/2)),x]

[Out]

(Sqrt[a + b*x]*Sqrt[c + d*x]*(3/b + (2*c^3)/((-(b*c) + a*d)*(c + d*x)^2) + (2*c^
2*(7*b*c - 9*a*d))/((b*c - a*d)^2*(c + d*x))))/(3*d^3) - ((5*b*c + a*d)*Log[b*c
+ a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(2*b^(3/2)*d^(
7/2))

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Maple [B]  time = 0.037, size = 928, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(d*x+c)^(5/2)/(b*x+a)^(1/2),x)

[Out]

-1/6*(b*x+a)^(1/2)*(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^2*a^3*d^5+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^2*b*c*d^4-27*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^2*c^2*d^3+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^3*c^3*d^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*a^3*
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^2*b*c^2*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))*x*a*b^2*c^3*d^2+30*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^3*c^4*d-6*x^2*a^2*d^4*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)+12*x^2*a*b*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*x^2*b^2*
c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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))*a^3*c^2*d^3+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^2*b*c^3*d^2-27*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^2*c^4*d+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))*b^3*
c^5-12*x*a^2*c*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+60*x*a*b*c^2*d^2*((b*x+a)
*(d*x+c))^(1/2)*(b*d)^(1/2)-40*x*b^2*c^3*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6
*a^2*c^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+44*a*b*c^3*d*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)-30*b^2*c^4*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2)/b/
(a*d-b*c)^2/((b*x+a)*(d*x+c))^(1/2)/d^3/(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^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.519775, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="fricas")

[Out]

[1/12*(4*(15*b^2*c^4 - 22*a*b*c^3*d + 3*a^2*c^2*d^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d
^3 + a^2*d^4)*x^2 + 2*(10*b^2*c^3*d - 15*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(b*d)
*sqrt(b*x + a)*sqrt(d*x + c) + 3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 +
a^3*c^2*d^3 + (5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 + a^3*d^5)*x^2 +
2*(5*b^3*c^4*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*d^4)*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^3*c^4*d^3 -
2*a*b^2*c^3*d^4 + a^2*b*c^2*d^5 + (b^3*c^2*d^5 - 2*a*b^2*c*d^6 + a^2*b*d^7)*x^2
+ 2*(b^3*c^3*d^4 - 2*a*b^2*c^2*d^5 + a^2*b*c*d^6)*x)*sqrt(b*d)), 1/6*(2*(15*b^2*
c^4 - 22*a*b*c^3*d + 3*a^2*c^2*d^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*x^2
+ 2*(10*b^2*c^3*d - 15*a*b*c^2*d^2 + 3*a^2*c*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*s
qrt(d*x + c) - 3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (5
*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(5*b^3*c^4*d -
9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*d^4)*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^3*c^4*d^3 - 2*a*b^2*c^3*d
^4 + a^2*b*c^2*d^5 + (b^3*c^2*d^5 - 2*a*b^2*c*d^6 + a^2*b*d^7)*x^2 + 2*(b^3*c^3*
d^4 - 2*a*b^2*c^2*d^5 + a^2*b*c*d^6)*x)*sqrt(-b*d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(d*x+c)**(5/2)/(b*x+a)**(1/2),x)

[Out]

Integral(x**3/(sqrt(a + b*x)*(c + d*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.25613, size = 504, normalized size = 2.9 \[ \frac{{\left ({\left (b x + a\right )}{\left (\frac{3 \,{\left (b^{6} c^{2} d^{4}{\left | b \right |} - 2 \, a b^{5} c d^{5}{\left | b \right |} + a^{2} b^{4} d^{6}{\left | b \right |}\right )}{\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac{2 \,{\left (10 \, b^{7} c^{3} d^{3}{\left | b \right |} - 18 \, a b^{6} c^{2} d^{4}{\left | b \right |} + 9 \, a^{2} b^{5} c d^{5}{\left | b \right |} - 3 \, a^{3} b^{4} d^{6}{\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac{3 \,{\left (5 \, b^{8} c^{4} d^{2}{\left | b \right |} - 14 \, a b^{7} c^{3} d^{3}{\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4}{\left | b \right |} - 4 \, a^{3} b^{5} c d^{5}{\left | b \right |} + a^{4} b^{4} d^{6}{\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt{b x + a}}{3 \,{\left (b^{2} c +{\left (b x + a\right )} b d - a b d\right )}^{\frac{3}{2}}} + \frac{{\left (5 \, b c{\left | b \right |} + a d{\left | b \right |}\right )}{\rm ln}\left ({\left | -\sqrt{b d} \sqrt{b x + a} + \sqrt{b^{2} c +{\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} b^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(sqrt(b*x + a)*(d*x + c)^(5/2)),x, algorithm="giac")

[Out]

1/3*((b*x + a)*(3*(b^6*c^2*d^4*abs(b) - 2*a*b^5*c*d^5*abs(b) + a^2*b^4*d^6*abs(b
))*(b*x + a)/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7) + 2*(10*b^7*c^3*d^3*abs
(b) - 18*a*b^6*c^2*d^4*abs(b) + 9*a^2*b^5*c*d^5*abs(b) - 3*a^3*b^4*d^6*abs(b))/(
b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)) + 3*(5*b^8*c^4*d^2*abs(b) - 14*a*b^7
*c^3*d^3*abs(b) + 12*a^2*b^6*c^2*d^4*abs(b) - 4*a^3*b^5*c*d^5*abs(b) + a^4*b^4*d
^6*abs(b))/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7))*sqrt(b*x + a)/(b^2*c + (
b*x + a)*b*d - a*b*d)^(3/2) + (5*b*c*abs(b) + a*d*abs(b))*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^2*d^3)