3.641 \(\int \frac{(a+b x)^{5/2} \sqrt{c+d x}}{x^4} \, dx\)

Optimal. Leaf size=228 \[ -\frac{\left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{5/2}}+2 b^{5/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d) (a d+b c)}{8 c^2 x}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+5 b c)}{12 c x^2} \]

[Out]

-((5*b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*c^2*x) - ((5*b*c + a
*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*c*x^2) - ((a + b*x)^(5/2)*Sqrt[c + d*x])/
(3*x^3) - ((5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[
c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(5/2)) + 2*b^(5/2)*Sqrt
[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 0.658065, antiderivative size = 228, 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 \[ -\frac{\left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 \sqrt{a} c^{5/2}}+2 b^{5/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} \sqrt{c+d x} (5 b c-a d) (a d+b c)}{8 c^2 x}-\frac{(a+b x)^{5/2} \sqrt{c+d x}}{3 x^3}-\frac{(a+b x)^{3/2} \sqrt{c+d x} (a d+5 b c)}{12 c x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*Sqrt[c + d*x])/x^4,x]

[Out]

-((5*b*c - a*d)*(b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*c^2*x) - ((5*b*c + a
*d)*(a + b*x)^(3/2)*Sqrt[c + d*x])/(12*c*x^2) - ((a + b*x)^(5/2)*Sqrt[c + d*x])/
(3*x^3) - ((5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[
c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(5/2)) + 2*b^(5/2)*Sqrt
[d]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi in Sympy [A]  time = 101.337, size = 212, normalized size = 0.93 \[ 2 b^{\frac{5}{2}} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + d x}}{\sqrt{d} \sqrt{a + b x}} \right )} - \frac{\left (a + b x\right )^{\frac{5}{2}} \sqrt{c + d x}}{3 x^{3}} - \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{c + d x} \left (a d + 5 b c\right )}{12 c x^{2}} + \frac{\sqrt{a + b x} \sqrt{c + d x} \left (a d - 5 b c\right ) \left (a d + b c\right )}{8 c^{2} x} - \frac{\left (a^{3} d^{3} - 5 a^{2} b c d^{2} + 15 a b^{2} c^{2} d + 5 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 \sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(1/2)/x**4,x)

[Out]

2*b**(5/2)*sqrt(d)*atanh(sqrt(b)*sqrt(c + d*x)/(sqrt(d)*sqrt(a + b*x))) - (a + b
*x)**(5/2)*sqrt(c + d*x)/(3*x**3) - (a + b*x)**(3/2)*sqrt(c + d*x)*(a*d + 5*b*c)
/(12*c*x**2) + sqrt(a + b*x)*sqrt(c + d*x)*(a*d - 5*b*c)*(a*d + b*c)/(8*c**2*x)
- (a**3*d**3 - 5*a**2*b*c*d**2 + 15*a*b**2*c**2*d + 5*b**3*c**3)*atanh(sqrt(c)*s
qrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*sqrt(a)*c**(5/2))

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Mathematica [A]  time = 0.517613, size = 283, normalized size = 1.24 \[ \sqrt{a+b x} \sqrt{c+d x} \left (\frac{3 a^2 d^2-14 a b c d-33 b^2 c^2}{24 c^2 x}-\frac{a^2}{3 x^3}-\frac{a (a d+13 b c)}{12 c x^2}\right )+\frac{\log (x) \left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right )}{16 \sqrt{a} c^{5/2}}-\frac{\left (a^3 d^3-5 a^2 b c d^2+15 a b^2 c^2 d+5 b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{16 \sqrt{a} c^{5/2}}+b^{5/2} \sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*Sqrt[c + d*x])/x^4,x]

[Out]

(-a^2/(3*x^3) - (a*(13*b*c + a*d))/(12*c*x^2) + (-33*b^2*c^2 - 14*a*b*c*d + 3*a^
2*d^2)/(24*c^2*x))*Sqrt[a + b*x]*Sqrt[c + d*x] + ((5*b^3*c^3 + 15*a*b^2*c^2*d -
5*a^2*b*c*d^2 + a^3*d^3)*Log[x])/(16*Sqrt[a]*c^(5/2)) - ((5*b^3*c^3 + 15*a*b^2*c
^2*d - 5*a^2*b*c*d^2 + a^3*d^3)*Log[2*a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sq
rt[a + b*x]*Sqrt[c + d*x]])/(16*Sqrt[a]*c^(5/2)) + b^(5/2)*Sqrt[d]*Log[b*c + a*d
+ 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[c + d*x]]

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Maple [B]  time = 0.025, size = 601, normalized size = 2.6 \[{\frac{1}{48\,{c}^{2}{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 48\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{3}{b}^{3}{c}^{2}d\sqrt{ac}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}\sqrt{bd}+15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}\sqrt{bd}-45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d\sqrt{bd}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}\sqrt{bd}+6\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}{a}^{2}\sqrt{ac}{x}^{2}-28\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}a\sqrt{ac}{x}^{2}c-66\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}\sqrt{ac}{x}^{2}-4\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{a}^{2}\sqrt{ac}xc-52\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}a\sqrt{ac}x-16\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^4,x)

[Out]

1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/c^2*(48*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^3*b^3*c^2*d*(a*c)^(1/2)-3*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^3*a^3*d^3*(b*
d)^(1/2)+15*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^3*a^2*b*c*d^2*(b*d)^(1/2)-45*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^3*a*b^2*c^2*d*(b*d)^(1/2)-15*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^3*b^3*c^3*(b*d)^(1/2)+6*(b*d
*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*(b*d)^(1/2)*a^2*(a*c)^(1/2)*x^2-28*(b*d*x^2+a*d*
x+b*c*x+a*c)^(1/2)*d*b*(b*d)^(1/2)*a*(a*c)^(1/2)*x^2*c-66*c^2*(b*d*x^2+a*d*x+b*c
*x+a*c)^(1/2)*b^2*(b*d)^(1/2)*(a*c)^(1/2)*x^2-4*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*
d*(b*d)^(1/2)*a^2*(a*c)^(1/2)*x*c-52*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b*(b*d)
^(1/2)*a*(a*c)^(1/2)*x-16*c^2*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*(b*d)^(1/2)*a^2*(a
*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/(b*d)^(1/2)/x^3/(a*c)^(1/2)

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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((b*x + a)^(5/2)*sqrt(d*x + c)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.51029, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x^4,x, algorithm="fricas")

[Out]

[1/96*(48*sqrt(a*c)*sqrt(b*d)*b^2*c^2*x^3*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*
d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*
(b^2*c*d + a*b*d^2)*x) + 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*c*d^2 + a^3*d^3
)*x^3*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c) -
(8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a^2*c*d)*x)*sqrt
(a*c))/x^2) - 4*(8*a^2*c^2 + (33*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*x^2 + 2*(13*a
*b*c^2 + a^2*c*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*c^2*x^3),
1/96*(96*sqrt(a*c)*sqrt(-b*d)*b^2*c^2*x^3*arctan(1/2*(2*b*d*x + b*c + a*d)/(sqr
t(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))) + 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b*
c*d^2 + a^3*d^3)*x^3*log(-(4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*s
qrt(d*x + c) - (8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 8*(a*b*c^2 + a
^2*c*d)*x)*sqrt(a*c))/x^2) - 4*(8*a^2*c^2 + (33*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2
)*x^2 + 2*(13*a*b*c^2 + a^2*c*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt
(a*c)*c^2*x^3), 1/48*(24*sqrt(-a*c)*sqrt(b*d)*b^2*c^2*x^3*log(8*b^2*d^2*x^2 + b^
2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sq
rt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) - 3*(5*b^3*c^3 + 15*a*b^2*c^2*d - 5*a^2*b
*c*d^2 + a^3*d^3)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)/(sqrt(b*x +
a)*sqrt(d*x + c)*a*c)) - 2*(8*a^2*c^2 + (33*b^2*c^2 + 14*a*b*c*d - 3*a^2*d^2)*x^
2 + 2*(13*a*b*c^2 + a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a
*c)*c^2*x^3), 1/48*(48*sqrt(-a*c)*sqrt(-b*d)*b^2*c^2*x^3*arctan(1/2*(2*b*d*x + b
*c + a*d)/(sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c))) - 3*(5*b^3*c^3 + 15*a*b^2*c^
2*d - 5*a^2*b*c*d^2 + a^3*d^3)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)
/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(8*a^2*c^2 + (33*b^2*c^2 + 14*a*b*c*d -
3*a^2*d^2)*x^2 + 2*(13*a*b*c^2 + a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x +
c))/(sqrt(-a*c)*c^2*x^3)]

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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((b*x+a)**(5/2)*(d*x+c)**(1/2)/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.686709, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*sqrt(d*x + c)/x^4,x, algorithm="giac")

[Out]

sage0*x