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

Optimal. Leaf size=198 \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2}}-\frac{c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{x}+\frac{3}{2} d \sqrt{a+b x} (c+d x)^{3/2}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 b} \]

[Out]

(d*(11*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b) + (3*d*Sqrt[a + b*x]*(c + d
*x)^(3/2))/2 - (Sqrt[a + b*x]*(c + d*x)^(5/2))/x - (c^(3/2)*(b*c + 5*a*d)*ArcTan
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] + (Sqrt[d]*(15*b^2*c
^2 + 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(4*b^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.657025, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )}{4 b^{3/2}}-\frac{c^{3/2} (5 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{\sqrt{a}}-\frac{\sqrt{a+b x} (c+d x)^{5/2}}{x}+\frac{3}{2} d \sqrt{a+b x} (c+d x)^{3/2}+\frac{d \sqrt{a+b x} \sqrt{c+d x} (a d+11 b c)}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

(d*(11*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(4*b) + (3*d*Sqrt[a + b*x]*(c + d
*x)^(3/2))/2 - (Sqrt[a + b*x]*(c + d*x)^(5/2))/x - (c^(3/2)*(b*c + 5*a*d)*ArcTan
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[a] + (Sqrt[d]*(15*b^2*c
^2 + 10*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x
])])/(4*b^(3/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 69.7545, size = 182, normalized size = 0.92 \[ \frac{3 d \sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}}}{2} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}}}{x} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (a d + 11 b c\right )}{4 b} - \frac{\sqrt{d} \left (a^{2} d^{2} - 10 a b c d - 15 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )}}{4 b^{\frac{3}{2}}} - \frac{c^{\frac{3}{2}} \left (5 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*d*sqrt(a + b*x)*(c + d*x)**(3/2)/2 - sqrt(a + b*x)*(c + d*x)**(5/2)/x + d*sqrt
(a + b*x)*sqrt(c + d*x)*(a*d + 11*b*c)/(4*b) - sqrt(d)*(a**2*d**2 - 10*a*b*c*d -
15*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d*x)))/(4*b**(3/2))
- c**(3/2)*(5*a*d + b*c)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/s
qrt(a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.553977, size = 214, normalized size = 1.08 \[ \frac{1}{8} \left (\frac{\sqrt{d} \left (-a^2 d^2+10 a b c d+15 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )}{b^{3/2}}+\frac{4 c^{3/2} \log (x) (5 a d+b c)}{\sqrt{a}}-\frac{4 c^{3/2} (5 a d+b c) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{\sqrt{a}}+2 \sqrt{a+b x} \sqrt{c+d x} \left (\frac{d^2 (a+2 b x)}{b}-\frac{4 c^2}{x}+9 c d\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*(9*c*d - (4*c^2)/x + (d^2*(a + 2*b*x))/b) + (4*c^
(3/2)*(b*c + 5*a*d)*Log[x])/Sqrt[a] - (4*c^(3/2)*(b*c + 5*a*d)*Log[2*a*c + b*c*x
+ a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/Sqrt[a] + (Sqrt[d]*(1
5*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sq
rt[a + b*x]*Sqrt[c + d*x]])/b^(3/2))/8

_______________________________________________________________________________________

Maple [B]  time = 0.023, size = 503, normalized size = 2.5 \[ -{\frac{1}{8\,bx}\sqrt{bx+a}\sqrt{dx+c} \left ( \ln \left ({\frac{1}{2} \left ( 2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc \right ){\frac{1}{\sqrt{bd}}}} \right ) x{a}^{2}{d}^{3}\sqrt{ac}-10\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) xabc{d}^{2}\sqrt{ac}-15\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ) x{b}^{2}{c}^{2}d\sqrt{ac}+20\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) xab{c}^{2}d\sqrt{bd}+4\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ) x{b}^{2}{c}^{3}\sqrt{bd}-4\,{x}^{2}b{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-2\,xa{d}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}-18\,xbcd\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+8\,b{c}^{2}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1
/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a^2*d^3*(a*c)^(1/2)-10*ln(1/2*(2*b*d*x+2
*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*x*a*b*c*d^2*(
a*c)^(1/2)-15*ln(1/2*(2*b*d*x+2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+a*d+
b*c)/(b*d)^(1/2))*x*b^2*c^2*d*(a*c)^(1/2)+20*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*(b*d*
x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x*a*b*c^2*d*(b*d)^(1/2)+4*ln((a*d*x+b*c*x+2
*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x*b^2*c^3*(b*d)^(1/2)-4*x
^2*b*d^2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-2*x*a*d^2*(a*c)
^(1/2)*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)-18*x*b*c*d*(a*c)^(1/2)*(b*d*x
^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)+8*b*c^2*(a*c)^(1/2)*(b*d*x^2+a*d*x+b*c*x+a
*c)^(1/2)*(b*d)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/b/(b*d)^(1/2)/x/(a*c)^(1/
2)

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 2.8966, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*((15*b^2*c^2 + 10*a*b*c*d - a^2*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*(b^2*c^2 + 5*a*b*c*d)*x*sqrt(c/a)
*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c + (a*b*c + a^
2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) -
4*(2*b*d^2*x^2 - 4*b*c^2 + (9*b*c*d + a*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*
x), 1/8*((15*b^2*c^2 + 10*a*b*c*d - a^2*d^2)*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x +
b*c + a*d)/(sqrt(b*x + a)*sqrt(d*x + c)*b*sqrt(-d/b))) + 2*(b^2*c^2 + 5*a*b*c*d)
*x*sqrt(c/a)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 - 4*(2*a^2*c +
(a*b*c + a^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(c/a) + 8*(a*b*c^2 + a^2*c*d
)*x)/x^2) + 2*(2*b*d^2*x^2 - 4*b*c^2 + (9*b*c*d + a*d^2)*x)*sqrt(b*x + a)*sqrt(d
*x + c))/(b*x), -1/16*(8*(b^2*c^2 + 5*a*b*c*d)*x*sqrt(-c/a)*arctan(1/2*(2*a*c +
(b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) + (15*b^2*c^2 + 10*a*
b*c*d - a^2*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*(2*b*d^2*x^2 - 4*b*c^2 + (9*b*c*d + a*d^2)*x)*sqrt(b*x + a)*
sqrt(d*x + c))/(b*x), -1/8*(4*(b^2*c^2 + 5*a*b*c*d)*x*sqrt(-c/a)*arctan(1/2*(2*a
*c + (b*c + a*d)*x)/(sqrt(b*x + a)*sqrt(d*x + c)*a*sqrt(-c/a))) - (15*b^2*c^2 +
10*a*b*c*d - a^2*d^2)*x*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqrt(b*x +
a)*sqrt(d*x + c)*b*sqrt(-d/b))) - 2*(2*b*d^2*x^2 - 4*b*c^2 + (9*b*c*d + a*d^2)*x
)*sqrt(b*x + a)*sqrt(d*x + c))/(b*x)]

_______________________________________________________________________________________

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((d*x+c)**(5/2)*(b*x+a)**(1/2)/x**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.596493, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x