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

Optimal. Leaf size=222 \[ \frac{5 \sqrt{d} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2}}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 b^4}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2} (3 b c-7 a d)}{6 b^3 (b c-a d)}-\frac{2 (c+d x)^{5/2} (3 b c-7 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)} \]

[Out]

(5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^4) + (5*d*(3*b*c - 7*a*d)
*Sqrt[a + b*x]*(c + d*x)^(3/2))/(6*b^3*(b*c - a*d)) - (2*(3*b*c - 7*a*d)*(c + d*
x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + (2*a*(c + d*x)^(7/2))/(3*b*(b*c -
a*d)*(a + b*x)^(3/2)) + (5*Sqrt[d]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.306934, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{5 \sqrt{d} (3 b c-7 a d) (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{9/2}}+\frac{5 d \sqrt{a+b x} \sqrt{c+d x} (3 b c-7 a d)}{4 b^4}+\frac{5 d \sqrt{a+b x} (c+d x)^{3/2} (3 b c-7 a d)}{6 b^3 (b c-a d)}-\frac{2 (c+d x)^{5/2} (3 b c-7 a d)}{3 b^2 \sqrt{a+b x} (b c-a d)}+\frac{2 a (c+d x)^{7/2}}{3 b (a+b x)^{3/2} (b c-a d)} \]

Antiderivative was successfully verified.

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

[Out]

(5*d*(3*b*c - 7*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b^4) + (5*d*(3*b*c - 7*a*d)
*Sqrt[a + b*x]*(c + d*x)^(3/2))/(6*b^3*(b*c - a*d)) - (2*(3*b*c - 7*a*d)*(c + d*
x)^(5/2))/(3*b^2*(b*c - a*d)*Sqrt[a + b*x]) + (2*a*(c + d*x)^(7/2))/(3*b*(b*c -
a*d)*(a + b*x)^(3/2)) + (5*Sqrt[d]*(3*b*c - 7*a*d)*(b*c - a*d)*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(4*b^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 35.3946, size = 207, normalized size = 0.93 \[ - \frac{2 a \left (c + d x\right )^{\frac{7}{2}}}{3 b \left (a + b x\right )^{\frac{3}{2}} \left (a d - b c\right )} - \frac{2 \left (c + d x\right )^{\frac{5}{2}} \left (7 a d - 3 b c\right )}{3 b^{2} \sqrt{a + b x} \left (a d - b c\right )} + \frac{5 d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (7 a d - 3 b c\right )}{6 b^{3} \left (a d - b c\right )} - \frac{5 d \sqrt{a + b x} \sqrt{c + d x} \left (7 a d - 3 b c\right )}{4 b^{4}} + \frac{5 \sqrt{d} \left (a d - b c\right ) \left (7 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*(c + d*x)**(7/2)/(3*b*(a + b*x)**(3/2)*(a*d - b*c)) - 2*(c + d*x)**(5/2)*(7
*a*d - 3*b*c)/(3*b**2*sqrt(a + b*x)*(a*d - b*c)) + 5*d*sqrt(a + b*x)*(c + d*x)**
(3/2)*(7*a*d - 3*b*c)/(6*b**3*(a*d - b*c)) - 5*d*sqrt(a + b*x)*sqrt(c + d*x)*(7*
a*d - 3*b*c)/(4*b**4) + 5*sqrt(d)*(a*d - b*c)*(7*a*d - 3*b*c)*atanh(sqrt(d)*sqrt
(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(4*b**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.276167, size = 173, normalized size = 0.78 \[ \frac{5 \sqrt{d} (3 b c-7 a d) (b c-a d) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{8 b^{9/2}}-\frac{\sqrt{c+d x} \left (105 a^3 d^2+5 a^2 b d (28 d x-23 c)+a b^2 \left (16 c^2-158 c d x+21 d^2 x^2\right )-3 b^3 x \left (-8 c^2+9 c d x+2 d^2 x^2\right )\right )}{12 b^4 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d*x]*(105*a^3*d^2 + 5*a^2*b*d*(-23*c + 28*d*x) - 3*b^3*x*(-8*c^2 + 9*
c*d*x + 2*d^2*x^2) + a*b^2*(16*c^2 - 158*c*d*x + 21*d^2*x^2)))/(12*b^4*(a + b*x)
^(3/2)) + (5*Sqrt[d]*(3*b*c - 7*a*d)*(b*c - a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqr
t[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(8*b^(9/2))

_______________________________________________________________________________________

Maple [B]  time = 0.034, size = 750, normalized size = 3.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(d*x+c)^(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*a^2*b^2*d^3-150*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*d^2+45*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^2*d+12*x^3*b^3*d^2*
((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+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*a^3*b*d^3-300*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+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*b^3*c^2*d-42*
x^2*a*b^2*d^2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+54*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^3-150*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+45*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-280*x*a^2*b*d^2*((b*x+a)*(
d*x+c))^(1/2)*(b*d)^(1/2)+316*x*a*b^2*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-48
*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)+230*a^2*b*c*d*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)-32*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)/(b*d)^(1/2)/(b*x+a)^(
3/2)/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)^(5/2)*x/(b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.708909, size = 1, normalized size = 0. \[ \left [\frac{15 \,{\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt{\frac{d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{d}{b}} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x\right ) + 4 \,{\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \,{\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \,{\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{48 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}, \frac{15 \,{\left (3 \, a^{2} b^{2} c^{2} - 10 \, a^{3} b c d + 7 \, a^{4} d^{2} +{\left (3 \, b^{4} c^{2} - 10 \, a b^{3} c d + 7 \, a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (3 \, a b^{3} c^{2} - 10 \, a^{2} b^{2} c d + 7 \, a^{3} b d^{2}\right )} x\right )} \sqrt{-\frac{d}{b}} \arctan \left (\frac{2 \, b d x + b c + a d}{2 \, \sqrt{b x + a} \sqrt{d x + c} b \sqrt{-\frac{d}{b}}}\right ) + 2 \,{\left (6 \, b^{3} d^{2} x^{3} - 16 \, a b^{2} c^{2} + 115 \, a^{2} b c d - 105 \, a^{3} d^{2} + 3 \,{\left (9 \, b^{3} c d - 7 \, a b^{2} d^{2}\right )} x^{2} - 2 \,{\left (12 \, b^{3} c^{2} - 79 \, a b^{2} c d + 70 \, a^{2} b d^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{24 \,{\left (b^{6} x^{2} + 2 \, a b^{5} x + a^{2} b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*(3*a^2*b^2*c^2 - 10*a^3*b*c*d + 7*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d
+ 7*a^2*b^2*d^2)*x^2 + 2*(3*a*b^3*c^2 - 10*a^2*b^2*c*d + 7*a^3*b*d^2)*x)*sqrt(d/
b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*
b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(6*b^3
*d^2*x^3 - 16*a*b^2*c^2 + 115*a^2*b*c*d - 105*a^3*d^2 + 3*(9*b^3*c*d - 7*a*b^2*d
^2)*x^2 - 2*(12*b^3*c^2 - 79*a*b^2*c*d + 70*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x
+ c))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), 1/24*(15*(3*a^2*b^2*c^2 - 10*a^3*b*c*d +
7*a^4*d^2 + (3*b^4*c^2 - 10*a*b^3*c*d + 7*a^2*b^2*d^2)*x^2 + 2*(3*a*b^3*c^2 - 1
0*a^2*b^2*c*d + 7*a^3*b*d^2)*x)*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqr
t(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) + 2*(6*b^3*d^2*x^3 - 16*a*b^2*c^2 + 115*
a^2*b*c*d - 105*a^3*d^2 + 3*(9*b^3*c*d - 7*a*b^2*d^2)*x^2 - 2*(12*b^3*c^2 - 79*a
*b^2*c*d + 70*a^2*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^6*x^2 + 2*a*b^5*x +
a^2*b^4)]

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.660931, 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)^(5/2)*x/(b*x + a)^(5/2),x, algorithm="giac")

[Out]

sage0*x