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

Optimal. Leaf size=246 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-\frac{35 a^2 d}{b}+10 a c+\frac{b c^2}{d}\right )}{12 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 \sqrt{a+b x} (b c-a d)}-\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{8 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.610439, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (-\frac{35 a^2 d}{b}+10 a c+\frac{b c^2}{d}\right )}{12 b^2 (b c-a d)}-\frac{2 a^2 (c+d x)^{5/2}}{b^2 \sqrt{a+b x} (b c-a d)}-\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{8 b^{9/2} d^{3/2}}-\frac{\sqrt{a+b x} \sqrt{c+d x} \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right )}{8 b^4 d}+\frac{\sqrt{a+b x} (c+d x)^{5/2}}{3 b^2 d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 44.1915, size = 228, normalized size = 0.93 \[ \frac{2 a^{2} \left (c + d x\right )^{\frac{5}{2}}}{b^{2} \sqrt{a + b x} \left (a d - b c\right )} + \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{3 b^{2} d} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (35 a^{2} d^{2} - 10 a b c d - b^{2} c^{2}\right )}{12 b^{3} d \left (a d - b c\right )} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (35 a^{2} d^{2} - 10 a b c d - b^{2} c^{2}\right )}{8 b^{4} d} - \frac{\left (a d - b c\right ) \left (35 a^{2} d^{2} - 10 a b c d - b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{8 b^{\frac{9}{2}} d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**2*(c + d*x)**(5/2)/(b**2*sqrt(a + b*x)*(a*d - b*c)) + sqrt(a + b*x)*(c + d*
x)**(5/2)/(3*b**2*d) - sqrt(a + b*x)*(c + d*x)**(3/2)*(35*a**2*d**2 - 10*a*b*c*d
- b**2*c**2)/(12*b**3*d*(a*d - b*c)) + sqrt(a + b*x)*sqrt(c + d*x)*(35*a**2*d**
2 - 10*a*b*c*d - b**2*c**2)/(8*b**4*d) - (a*d - b*c)*(35*a**2*d**2 - 10*a*b*c*d
- b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(8*b**(9/2)*d*
*(3/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.18973, size = 188, normalized size = 0.76 \[ \frac{\sqrt{c+d x} \left (105 a^3 d^2+5 a^2 b d (7 d x-20 c)+a b^2 \left (3 c^2-38 c d x-14 d^2 x^2\right )+b^3 x \left (3 c^2+14 c d x+8 d^2 x^2\right )\right )}{24 b^4 d \sqrt{a+b x}}-\frac{(b c-a d) \left (-35 a^2 d^2+10 a b c d+b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{16 b^{9/2} d^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c + d*x]*(105*a^3*d^2 + 5*a^2*b*d*(-20*c + 7*d*x) + a*b^2*(3*c^2 - 38*c*d*
x - 14*d^2*x^2) + b^3*x*(3*c^2 + 14*c*d*x + 8*d^2*x^2)))/(24*b^4*d*Sqrt[a + b*x]
) - ((b*c - a*d)*(b^2*c^2 + 10*a*b*c*d - 35*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]])/(16*b^(9/2)*d^(3/2))

_______________________________________________________________________________________

Maple [B]  time = 0.038, size = 692, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(d*x+c)^(1/2)*(-16*x^3*b^3*d^2*((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*a^3*b
*d^3-135*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*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))*x*a*b^3*c^2*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^4*c^3+28*x^2*a*b^2*d^2*((b*x+a)*(d*x+c))^(1/
2)*(b*d)^(1/2)-28*x^2*b^3*c*d*((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))*a^4*d^3-135*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*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^2*b^2*c^2*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^3*c^3-70*x*a^2*b*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+76
*x*a*b^2*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*x*b^3*c^2*((b*x+a)*(d*x+c))^(
1/2)*(b*d)^(1/2)-210*a^3*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+200*a^2*b*c*d*(
(b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-6*a*b^2*c^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/
2))/((b*x+a)*(d*x+c))^(1/2)/d/(b*d)^(1/2)/(b*x+a)^(1/2)/b^4

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.59094, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (8 \, b^{3} d^{2} x^{3} + 3 \, a b^{2} c^{2} - 100 \, a^{2} b c d + 105 \, a^{3} d^{2} + 14 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} +{\left (3 \, b^{3} c^{2} - 38 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b d} \sqrt{b x + a} \sqrt{d x + c} + 3 \,{\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\right )} \log \left (-4 \,{\left (2 \, b^{2} d^{2} x + b^{2} c d + a b d^{2}\right )} \sqrt{b x + a} \sqrt{d x + c} +{\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right )} \sqrt{b d}\right )}{96 \,{\left (b^{5} d x + a b^{4} d\right )} \sqrt{b d}}, \frac{2 \,{\left (8 \, b^{3} d^{2} x^{3} + 3 \, a b^{2} c^{2} - 100 \, a^{2} b c d + 105 \, a^{3} d^{2} + 14 \,{\left (b^{3} c d - a b^{2} d^{2}\right )} x^{2} +{\left (3 \, b^{3} c^{2} - 38 \, a b^{2} c d + 35 \, a^{2} b d^{2}\right )} x\right )} \sqrt{-b d} \sqrt{b x + a} \sqrt{d x + c} - 3 \,{\left (a b^{3} c^{3} + 9 \, a^{2} b^{2} c^{2} d - 45 \, a^{3} b c d^{2} + 35 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 9 \, a b^{3} c^{2} d - 45 \, a^{2} b^{2} c d^{2} + 35 \, a^{3} b d^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x + a} \sqrt{d x + c} b d}\right )}{48 \,{\left (b^{5} d x + a b^{4} d\right )} \sqrt{-b d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*b^3*d^2*x^3 + 3*a*b^2*c^2 - 100*a^2*b*c*d + 105*a^3*d^2 + 14*(b^3*c*
d - a*b^2*d^2)*x^2 + (3*b^3*c^2 - 38*a*b^2*c*d + 35*a^2*b*d^2)*x)*sqrt(b*d)*sqrt
(b*x + a)*sqrt(d*x + c) + 3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d - 45*a^3*b*c*d^2 + 35*a
^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*a^3*b*d^3)*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^5*d*x +
a*b^4*d)*sqrt(b*d)), 1/48*(2*(8*b^3*d^2*x^3 + 3*a*b^2*c^2 - 100*a^2*b*c*d + 105
*a^3*d^2 + 14*(b^3*c*d - a*b^2*d^2)*x^2 + (3*b^3*c^2 - 38*a*b^2*c*d + 35*a^2*b*d
^2)*x)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c) - 3*(a*b^3*c^3 + 9*a^2*b^2*c^2*d -
45*a^3*b*c*d^2 + 35*a^4*d^3 + (b^4*c^3 + 9*a*b^3*c^2*d - 45*a^2*b^2*c*d^2 + 35*
a^3*b*d^3)*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^5*d*x + a*b^4*d)*sqrt(-b*d))]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.638716, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x