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

Optimal. Leaf size=316 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{64 a^2 c x}-\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 b^2 c}{a}-\frac{5 a d^2}{c}+50 b d\right )}{96 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{24 c x^3} \]

[Out]

((3*b^3*c^3 - 17*a*b^2*c^2*d - 55*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a^2*c*x) - (((3*b^2*c)/a + 50*b*d - (5*a*d^2)/c)*Sqrt[a + b*x]*(c +
d*x)^(3/2))/(96*x^2) - ((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*c*x^3
) - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*x^4) - ((3*b^4*c^4 - 20*a*b^3*c^3*d + 9
0*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(3/2)) + 2*b^(3/2)*d^(5/2)*ArcTanh[(Sqrt
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]

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Rubi [A]  time = 0.979677, antiderivative size = 316, 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 \[ \frac{\sqrt{a+b x} \sqrt{c+d x} \left (5 a^3 d^3-55 a^2 b c d^2-17 a b^2 c^2 d+3 b^3 c^3\right )}{64 a^2 c x}-\frac{\left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{64 a^{5/2} c^{3/2}}+2 b^{3/2} d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 b^2 c}{a}-\frac{5 a d^2}{c}+50 b d\right )}{96 x^2}-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{4 x^4}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{24 c x^3} \]

Antiderivative was successfully verified.

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

[Out]

((3*b^3*c^3 - 17*a*b^2*c^2*d - 55*a^2*b*c*d^2 + 5*a^3*d^3)*Sqrt[a + b*x]*Sqrt[c
+ d*x])/(64*a^2*c*x) - (((3*b^2*c)/a + 50*b*d - (5*a*d^2)/c)*Sqrt[a + b*x]*(c +
d*x)^(3/2))/(96*x^2) - ((3*b*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(24*c*x^3
) - ((a + b*x)^(3/2)*(c + d*x)^(5/2))/(4*x^4) - ((3*b^4*c^4 - 20*a*b^3*c^3*d + 9
0*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/
(Sqrt[a]*Sqrt[c + d*x])])/(64*a^(5/2)*c^(3/2)) + 2*b^(3/2)*d^(5/2)*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)**(3/2)*(d*x+c)**(5/2)/x**5,x)

[Out]

Timed out

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Mathematica [A]  time = 0.319529, size = 353, normalized size = 1.12 \[ \frac{1}{384} \left (-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^3 \left (48 c^3+136 c^2 d x+118 c d^2 x^2+15 d^3 x^3\right )+a^2 b c x \left (72 c^2+244 c d x+337 d^2 x^2\right )+3 a b^2 c^2 x^2 (2 c+19 d x)-9 b^3 c^3 x^3\right )}{a^2 c x^4}+\frac{3 \log (x) \left (-5 a^4 d^4+60 a^3 b c d^3+90 a^2 b^2 c^2 d^2-20 a b^3 c^3 d+3 b^4 c^4\right )}{a^{5/2} c^{3/2}}+\frac{3 \left (5 a^4 d^4-60 a^3 b c d^3-90 a^2 b^2 c^2 d^2+20 a b^3 c^3 d-3 b^4 c^4\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{5/2} c^{3/2}}+384 b^{3/2} d^{5/2} \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(-9*b^3*c^3*x^3 + 3*a*b^2*c^2*x^2*(2*c + 19*d*x
) + a^2*b*c*x*(72*c^2 + 244*c*d*x + 337*d^2*x^2) + a^3*(48*c^3 + 136*c^2*d*x + 1
18*c*d^2*x^2 + 15*d^3*x^3)))/(a^2*c*x^4) + (3*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a
^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*Log[x])/(a^(5/2)*c^(3/2)) + (3*(-3*
b^4*c^4 + 20*a*b^3*c^3*d - 90*a^2*b^2*c^2*d^2 - 60*a^3*b*c*d^3 + 5*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^(5/2)
*c^(3/2)) + 384*b^(3/2)*d^(5/2)*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 = 0.029, size = 852, normalized size = 2.7 \[{\frac{1}{384\,{a}^{2}c{x}^{4}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{4}{d}^{4}\sqrt{bd}-180\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{3}bc{d}^{3}\sqrt{bd}-270\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{a}^{2}{b}^{2}{c}^{2}{d}^{2}\sqrt{bd}+60\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}a{b}^{3}{c}^{3}d\sqrt{bd}-9\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{4}{b}^{4}{c}^{4}\sqrt{bd}+384\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{4}{a}^{2}{b}^{2}c{d}^{3}\sqrt{ac}-30\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{3}\sqrt{bd}{a}^{3}{x}^{3}\sqrt{ac}-674\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}b\sqrt{bd}c{a}^{2}{x}^{3}\sqrt{ac}-114\,\sqrt{d{x}^{2}b+adx+bcx+ac}d{b}^{2}\sqrt{bd}{c}^{2}a{x}^{3}\sqrt{ac}+18\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{3}\sqrt{bd}{x}^{3}\sqrt{ac}-236\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}c{a}^{3}{x}^{2}\sqrt{ac}-488\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}{c}^{2}{a}^{2}{x}^{2}\sqrt{ac}-12\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}a{x}^{2}\sqrt{ac}-272\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{c}^{2}{a}^{3}x\sqrt{ac}-144\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}{a}^{2}x\sqrt{ac}-96\,{c}^{3}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{3}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 6.3078, 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)^(3/2)*(d*x + c)^(5/2)/x^5,x, algorithm="fricas")

[Out]

[1/768*(384*sqrt(a*c)*sqrt(b*d)*a^2*b*c*d^2*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*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^
2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*x^4*log((4*(2*a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)*s
qrt(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 - (9*b^3*c^3 - 57*a*b^2
*c^2*d - 337*a^2*b*c*d^2 - 15*a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 122*a^2*b*c^2*d +
59*a^3*c*d^2)*x^2 + 8*(9*a^2*b*c^3 + 17*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sq
rt(d*x + c))/(sqrt(a*c)*a^2*c*x^4), 1/768*(768*sqrt(a*c)*sqrt(-b*d)*a^2*b*c*d^2*
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*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*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 - (9*b^3*c^3 - 57*a*b^2*c^2*d - 337*a^2*b*c*d^2 - 15*
a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 122*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 + 8*(9*a^2*b
*c^3 + 17*a^3*c^2*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a^2*c*
x^4), 1/384*(192*sqrt(-a*c)*sqrt(b*d)*a^2*b*c*d^2*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*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*
c^2*d^2 + 60*a^3*b*c*d^3 - 5*a^4*d^4)*x^4*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqr
t(-a*c)/(sqrt(b*x + a)*sqrt(d*x + c)*a*c)) - 2*(48*a^3*c^3 - (9*b^3*c^3 - 57*a*b
^2*c^2*d - 337*a^2*b*c*d^2 - 15*a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 122*a^2*b*c^2*d
+ 59*a^3*c*d^2)*x^2 + 8*(9*a^2*b*c^3 + 17*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)
*sqrt(d*x + c))/(sqrt(-a*c)*a^2*c*x^4), 1/384*(384*sqrt(-a*c)*sqrt(-b*d)*a^2*b*c
*d^2*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*(3*b^4*c^4 - 20*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 + 60*a^3*b*c*d^3 - 5*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 - (9*b^3*c^3 - 57*a*b^2*c^2*d - 337*a^2*b*c*d^2 - 15
*a^3*d^3)*x^3 + 2*(3*a*b^2*c^3 + 122*a^2*b*c^2*d + 59*a^3*c*d^2)*x^2 + 8*(9*a^2*
b*c^3 + 17*a^3*c^2*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a^2
*c*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)**(3/2)*(d*x+c)**(5/2)/x**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.702767, 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)^(3/2)*(d*x + c)^(5/2)/x^5,x, algorithm="giac")

[Out]

sage0*x