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

Optimal. Leaf size=271 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{32 b^4 d^2}+\frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{9 x^2 \sqrt{a+b x} (c+d x)^{3/2}}{4 b^2}-\frac{2 x^3 (c+d x)^{3/2}}{b \sqrt{a+b x}} \]

[Out]

(3*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*b^5*d^2) - (2*x^3*(c + d*x)^(3/2))/(b*Sqrt[a + b*x]) + (9*x^2*Sqrt[
a + b*x]*(c + d*x)^(3/2))/(4*b^2) - (Sqrt[a + b*x]*(c + d*x)^(3/2)*(3*b^2*c^2 +
14*a*b*c*d - 105*a^2*d^2 - 4*b*d*(b*c - 21*a*d)*x))/(32*b^4*d^2) + (3*(b*c - a*d
)*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt
[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(11/2)*d^(5/2))

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Rubi [A]  time = 0.586291, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{32 b^4 d^2}+\frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{64 b^{11/2} d^{5/2}}+\frac{3 \sqrt{a+b x} \sqrt{c+d x} \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right )}{64 b^5 d^2}+\frac{9 x^2 \sqrt{a+b x} (c+d x)^{3/2}}{4 b^2}-\frac{2 x^3 (c+d x)^{3/2}}{b \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*b^5*d^2) - (2*x^3*(c + d*x)^(3/2))/(b*Sqrt[a + b*x]) + (9*x^2*Sqrt[
a + b*x]*(c + d*x)^(3/2))/(4*b^2) - (Sqrt[a + b*x]*(c + d*x)^(3/2)*(3*b^2*c^2 +
14*a*b*c*d - 105*a^2*d^2 - 4*b*d*(b*c - 21*a*d)*x))/(32*b^4*d^2) + (3*(b*c - a*d
)*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*ArcTanh[(Sqrt[d]*Sqrt
[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(64*b^(11/2)*d^(5/2))

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Rubi in Sympy [A]  time = 52.4902, size = 277, normalized size = 1.02 \[ - \frac{2 x^{3} \left (c + d x\right )^{\frac{3}{2}}}{b \sqrt{a + b x}} + \frac{9 x^{2} \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{4 b^{2}} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (\frac{315 a^{2} d^{2}}{8} - \frac{21 a b c d}{4} - \frac{9 b^{2} c^{2}}{8} - \frac{3 b d x \left (21 a d - b c\right )}{2}\right )}{12 b^{4} d^{2}} - \frac{3 \sqrt{a + b x} \sqrt{c + d x} \left (105 a^{3} d^{3} - 35 a^{2} b c d^{2} - 5 a b^{2} c^{2} d - b^{3} c^{3}\right )}{64 b^{5} d^{2}} + \frac{3 \left (a d - b c\right ) \left (105 a^{3} d^{3} - 35 a^{2} b c d^{2} - 5 a b^{2} c^{2} d - b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{64 b^{\frac{11}{2}} d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**3*(c + d*x)**(3/2)/(b*sqrt(a + b*x)) + 9*x**2*sqrt(a + b*x)*(c + d*x)**(3/
2)/(4*b**2) + sqrt(a + b*x)*(c + d*x)**(3/2)*(315*a**2*d**2/8 - 21*a*b*c*d/4 - 9
*b**2*c**2/8 - 3*b*d*x*(21*a*d - b*c)/2)/(12*b**4*d**2) - 3*sqrt(a + b*x)*sqrt(c
+ d*x)*(105*a**3*d**3 - 35*a**2*b*c*d**2 - 5*a*b**2*c**2*d - b**3*c**3)/(64*b**
5*d**2) + 3*(a*d - b*c)*(105*a**3*d**3 - 35*a**2*b*c*d**2 - 5*a*b**2*c**2*d - b*
*3*c**3)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(64*b**(11/2)*d**(
5/2))

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Mathematica [A]  time = 0.264414, size = 253, normalized size = 0.93 \[ \frac{3 (b c-a d) \left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{128 b^{11/2} d^{5/2}}-\frac{\sqrt{c+d x} \left (315 a^4 d^3+105 a^3 b d^2 (d x-3 c)+a^2 b^2 d \left (13 c^2-119 c d x-42 d^2 x^2\right )+a b^3 \left (3 c^3+11 c^2 d x+44 c d^2 x^2+24 d^3 x^3\right )-b^4 x \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )\right )}{64 b^5 d^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(315*a^4*d^3 + 105*a^3*b*d^2*(-3*c + d*x) + a^2*b^2*d*(13*c^2 -
119*c*d*x - 42*d^2*x^2) - b^4*x*(-3*c^3 + 2*c^2*d*x + 24*c*d^2*x^2 + 16*d^3*x^3)
+ a*b^3*(3*c^3 + 11*c^2*d*x + 44*c*d^2*x^2 + 24*d^3*x^3)))/(64*b^5*d^2*Sqrt[a +
b*x]) + (3*(b*c - a*d)*(b^3*c^3 + 5*a*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)
*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(128*
b^(11/2)*d^(5/2))

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Maple [B]  time = 0.047, size = 961, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(d*x+c)^(1/2)*(32*x^4*b^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-48*x^3*a
*b^3*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+48*x^3*b^4*c*d^2*((b*x+a)*(d*x+c))^
(1/2)*(b*d)^(1/2)+315*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*d^4-420*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*d^3+90*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^2*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))*x*a*b^4*c^3*d+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*b^5*
c^4+84*x^2*a^2*b^2*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-88*x^2*a*b^3*c*d^2*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+4*x^2*b^4*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^
(1/2)+315*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*d^4-420*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*d^3+90*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^2*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^3*d+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*b^4*c^4-210*x*a^3*b*d^3*((
b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+238*x*a^2*b^2*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b
*d)^(1/2)-22*x*a*b^3*c^2*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*x*b^4*c^3*((b*x
+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-630*a^4*d^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+6
30*a^3*b*c*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-26*a^2*b^2*c^2*d*((b*x+a)*(d*
x+c))^(1/2)*(b*d)^(1/2)-6*a*b^3*c^3*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/((b*x+a
)*(d*x+c))^(1/2)/(b*d)^(1/2)/d^2/(b*x+a)^(1/2)/b^5

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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((d*x + c)^(3/2)*x^3/(b*x + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.981152, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (16 \, b^{4} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 13 \, a^{2} b^{2} c^{2} d + 315 \, a^{3} b c d^{2} - 315 \, a^{4} d^{3} + 24 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (b^{4} c^{2} d - 22 \, a b^{3} c d^{2} + 21 \, a^{2} b^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{4} c^{3} + 11 \, a b^{3} c^{2} d - 119 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{256 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )} \sqrt{b d}}, \frac{2 \,{\left (16 \, b^{4} d^{3} x^{4} - 3 \, a b^{3} c^{3} - 13 \, a^{2} b^{2} c^{2} d + 315 \, a^{3} b c d^{2} - 315 \, a^{4} d^{3} + 24 \,{\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 2 \,{\left (b^{4} c^{2} d - 22 \, a b^{3} c d^{2} + 21 \, a^{2} b^{2} d^{3}\right )} x^{2} -{\left (3 \, b^{4} c^{3} + 11 \, a b^{3} c^{2} d - 119 \, a^{2} b^{2} c d^{2} + 105 \, a^{3} b d^{3}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d + 30 \, a^{3} b^{2} c^{2} d^{2} - 140 \, a^{4} b c d^{3} + 105 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d + 30 \, a^{2} b^{3} c^{2} d^{2} - 140 \, a^{3} b^{2} c d^{3} + 105 \, a^{4} b d^{4}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{128 \,{\left (b^{6} d^{2} x + a b^{5} d^{2}\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/256*(4*(16*b^4*d^3*x^4 - 3*a*b^3*c^3 - 13*a^2*b^2*c^2*d + 315*a^3*b*c*d^2 - 3
15*a^4*d^3 + 24*(b^4*c*d^2 - a*b^3*d^3)*x^3 + 2*(b^4*c^2*d - 22*a*b^3*c*d^2 + 21
*a^2*b^2*d^3)*x^2 - (3*b^4*c^3 + 11*a*b^3*c^2*d - 119*a^2*b^2*c*d^2 + 105*a^3*b*
d^3)*x)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(a*b^4*c^4 + 4*a^2*b^3*c^3*d +
30*a^3*b^2*c^2*d^2 - 140*a^4*b*c*d^3 + 105*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d +
30*a^2*b^3*c^2*d^2 - 140*a^3*b^2*c*d^3 + 105*a^4*b*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^6*d^2*x + a*b^5*d^2)
*sqrt(b*d)), 1/128*(2*(16*b^4*d^3*x^4 - 3*a*b^3*c^3 - 13*a^2*b^2*c^2*d + 315*a^3
*b*c*d^2 - 315*a^4*d^3 + 24*(b^4*c*d^2 - a*b^3*d^3)*x^3 + 2*(b^4*c^2*d - 22*a*b^
3*c*d^2 + 21*a^2*b^2*d^3)*x^2 - (3*b^4*c^3 + 11*a*b^3*c^2*d - 119*a^2*b^2*c*d^2
+ 105*a^3*b*d^3)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 3*(a*b^4*c^4 + 4*a^
2*b^3*c^3*d + 30*a^3*b^2*c^2*d^2 - 140*a^4*b*c*d^3 + 105*a^5*d^4 + (b^5*c^4 + 4*
a*b^4*c^3*d + 30*a^2*b^3*c^2*d^2 - 140*a^3*b^2*c*d^3 + 105*a^4*b*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^6*d
^2*x + a*b^5*d^2)*sqrt(-b*d))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.618315, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x