3.651 \(\int \frac{(a+b x)^{5/2} (c+d x)^{3/2}}{x^5} \, dx\)

Optimal. Leaf size=313 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac{\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \]

[Out]

-((5*b^3*c^3 + 73*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a*c^2*x) - ((5*b*c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/
2))/(32*c^2*x^2) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*c*x^3)
- ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*x^4) + ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*
a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5/2)) + 2*b^(5/2)*d^(3/2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 1.02204, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{\sqrt{a+b x} \sqrt{c+d x} \left (3 a^3 d^3-17 a^2 b c d^2+73 a b^2 c^2 d+5 b^3 c^3\right )}{64 a c^2 x}+\frac{\left (-3 a^4 d^4+20 a^3 b c d^3-90 a^2 b^2 c^2 d^2-60 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{3/2} c^{5/2}}+2 b^{5/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 b c-a d) (3 a d+b c)}{32 c^2 x^2}-\frac{(a+b x)^{5/2} (c+d x)^{3/2}}{4 x^4}-\frac{(a+b x)^{3/2} (c+d x)^{3/2} (3 a d+5 b c)}{24 c x^3} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^5,x]

[Out]

-((5*b^3*c^3 + 73*a*b^2*c^2*d - 17*a^2*b*c*d^2 + 3*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a*c^2*x) - ((5*b*c - a*d)*(b*c + 3*a*d)*Sqrt[a + b*x]*(c + d*x)^(3/
2))/(32*c^2*x^2) - ((5*b*c + 3*a*d)*(a + b*x)^(3/2)*(c + d*x)^(3/2))/(24*c*x^3)
- ((a + b*x)^(5/2)*(c + d*x)^(3/2))/(4*x^4) + ((5*b^4*c^4 - 60*a*b^3*c^3*d - 90*
a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(S
qrt[a]*Sqrt[c + d*x])])/(64*a^(3/2)*c^(5/2)) + 2*b^(5/2)*d^(3/2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(5/2)*(d*x+c)**(3/2)/x**5,x)

[Out]

Timed out

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Mathematica [A]  time = 0.37009, size = 352, normalized size = 1.12 \[ \frac{1}{384} \left (-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+72 c^2 d x+6 c d^2 x^2-9 d^3 x^3\right )+a^2 b c x \left (136 c^2+244 c d x+57 d^2 x^2\right )+a b^2 c^2 x^2 (118 c+337 d x)+15 b^3 c^3 x^3\right )}{a c^2 x^4}+\frac{3 \log (x) \left (3 a^4 d^4-20 a^3 b c d^3+90 a^2 b^2 c^2 d^2+60 a b^3 c^3 d-5 b^4 c^4\right )}{a^{3/2} c^{5/2}}-\frac{3 \left (3 a^4 d^4-20 a^3 b c d^3+90 a^2 b^2 c^2 d^2+60 a b^3 c^3 d-5 b^4 c^4\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2} c^{5/2}}+384 b^{5/2} d^{3/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(5/2)*(c + d*x)^(3/2))/x^5,x]

[Out]

((-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 + a*b^2*c^2*x^2*(118*c + 337*d*
x) + a^2*b*c*x*(136*c^2 + 244*c*d*x + 57*d^2*x^2) + a^3*(48*c^3 + 72*c^2*d*x + 6
*c*d^2*x^2 - 9*d^3*x^3)))/(a*c^2*x^4) + (3*(-5*b^4*c^4 + 60*a*b^3*c^3*d + 90*a^2
*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*Log[x])/(a^(3/2)*c^(5/2)) - (3*(-5*b^
4*c^4 + 60*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 3*a^4*d^4)*Log[2*
a*c + b*c*x + a*d*x + 2*Sqrt[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(a^(3/2)*c
^(5/2)) + 384*b^(5/2)*d^(3/2)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a
+ b*x]*Sqrt[c + d*x]])/384

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Maple [B]  time = 0.027, size = 852, normalized size = 2.7 \[ -{\frac{1}{384\,a{c}^{2}{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}\sqrt{bd}-60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}\sqrt{bd}+270\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}\sqrt{bd}+180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d\sqrt{bd}-15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}\sqrt{bd}-384\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}a{b}^{3}{c}^{2}{d}^{2}\sqrt{ac}-18\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{3}\sqrt{bd}{a}^{3}{x}^{3}\sqrt{ac}+114\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}b\sqrt{bd}c{a}^{2}{x}^{3}\sqrt{ac}+674\,\sqrt{d{x}^{2}b+adx+bcx+ac}d{b}^{2}\sqrt{bd}{c}^{2}a{x}^{3}\sqrt{ac}+30\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}\sqrt{bd}{x}^{3}\sqrt{ac}+12\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}c{a}^{3}{x}^{2}\sqrt{ac}+488\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}+236\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}a{x}^{2}\sqrt{ac}+144\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{c}^{2}{a}^{3}x\sqrt{ac}+272\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}{a}^{2}x\sqrt{ac}+96\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(5/2)*(d*x+c)^(3/2)/x^5,x)

[Out]

-1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c^2*(9*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^4*a^4*d^4*(b*d)^(1/2)-60*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^4*a^3*b*c*d^3*(b*d)^(1/
2)+270*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
^4*a^2*b^2*c^2*d^2*(b*d)^(1/2)+180*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^4*a*b^3*c^3*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^4*b^4*c^4*(b*d)^(1/2)-384*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^4*a*b^3*c^2*d^2*(a*c)^(1/2)-18*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^3*(b*d)^(1/
2)*a^3*x^3*(a*c)^(1/2)+114*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*b*(b*d)^(1/2)*c*a
^2*x^3*(a*c)^(1/2)+674*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b^2*(b*d)^(1/2)*c^2*a*x
^3*(a*c)^(1/2)+30*c^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^3*(b*d)^(1/2)*x^3*(a*c)^
(1/2)+12*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d^2*(b*d)^(1/2)*c*a^3*x^2*(a*c)^(1/2)+4
88*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*d*b*(b*d)^(1/2)*c^2*a^2*x^2*(a*c)^(1/2)+236*c
^3*(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)*b^2*(b*d)^(1/2)*a*x^2*(a*c)^(1/2)+144*(b*d*x^
2+a*d*x+b*c*x+a*c)^(1/2)*d*(b*d)^(1/2)*c^2*a^3*x*(a*c)^(1/2)+272*c^3*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)*b*(b*d)^(1/2)*a^2*x*(a*c)^(1/2)+96*c^3*(b*d*x^2+a*d*x+b*c*x
+a*c)^(1/2)*(b*d)^(1/2)*a^3*(a*c)^(1/2))/(b*d*x^2+a*d*x+b*c*x+a*c)^(1/2)/x^4/(a*
c)^(1/2)/(b*d)^(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)*(d*x + c)^(3/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.59461, 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)*(d*x + c)^(3/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(384*sqrt(a*c)*sqrt(b*d)*a*b^2*c^2*d*x^4*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^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^
2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*x^4*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*(48*a^3*c^3 + (15*b^3*c^3 + 337*a*
b^2*c^2*d + 57*a^2*b*c*d^2 - 9*a^3*d^3)*x^3 + 2*(59*a*b^2*c^3 + 122*a^2*b*c^2*d
+ 3*a^3*c*d^2)*x^2 + 8*(17*a^2*b*c^3 + 9*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*s
qrt(d*x + c))/(sqrt(a*c)*a*c^2*x^4), 1/768*(768*sqrt(a*c)*sqrt(-b*d)*a*b^2*c^2*d
*x^4*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^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^
4)*x^4*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)*sqr
t(a*c))/x^2) - 4*(48*a^3*c^3 + (15*b^3*c^3 + 337*a*b^2*c^2*d + 57*a^2*b*c*d^2 -
9*a^3*d^3)*x^3 + 2*(59*a*b^2*c^3 + 122*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 + 8*(17*a^
2*b*c^3 + 9*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*c^
2*x^4), 1/384*(192*sqrt(-a*c)*sqrt(b*d)*a*b^2*c^2*d*x^4*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^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^
2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*a^4*d^4)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*s
qrt(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(48*a^3*c^3 + (15*b^3*c^3 + 337
*a*b^2*c^2*d + 57*a^2*b*c*d^2 - 9*a^3*d^3)*x^3 + 2*(59*a*b^2*c^3 + 122*a^2*b*c^2
*d + 3*a^3*c*d^2)*x^2 + 8*(17*a^2*b*c^3 + 9*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x +
a)*sqrt(d*x + c))/(sqrt(-a*c)*a*c^2*x^4), 1/384*(384*sqrt(-a*c)*sqrt(-b*d)*a*b^2
*c^2*d*x^4*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^4*c^4 - 60*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 3*
a^4*d^4)*x^4*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*(48*a^3*c^3 + (15*b^3*c^3 + 337*a*b^2*c^2*d + 57*a^2*b*c*d^2 -
9*a^3*d^3)*x^3 + 2*(59*a*b^2*c^3 + 122*a^2*b*c^2*d + 3*a^3*c*d^2)*x^2 + 8*(17*a
^2*b*c^3 + 9*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a
*c^2*x^4)]

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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)**(3/2)/x**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.693884, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^(5/2)*(d*x + c)^(3/2)/x^5,x, algorithm="giac")

[Out]

sage0*x