3.610 \(\int \frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x^4} \, dx\)

Optimal. Leaf size=292 \[ \frac{d \sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+26 a b c d+b^2 c^2\right )}{8 a c}+\frac{\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 b^2 c}{a}+\frac{5 a d^2}{c}+40 b d\right )}{24 x}+\sqrt{b} d^{3/2} (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{12 c x^2} \]

[Out]

(d*(b^2*c^2 + 26*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a*c) - (((
3*b^2*c)/a + 40*b*d + (5*a*d^2)/c)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(24*x) - ((3*b
*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(12*c*x^2) - ((a + b*x)^(3/2)*(c + d*
x)^(5/2))/(3*x^3) + ((b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*Sqrt[c]) + Sqr
t[b]*d^(3/2)*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d
*x])]

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Rubi [A]  time = 1.00782, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{d \sqrt{a+b x} \sqrt{c+d x} \left (5 a^2 d^2+26 a b c d+b^2 c^2\right )}{8 a c}+\frac{\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{8 a^{3/2} \sqrt{c}}-\frac{\sqrt{a+b x} (c+d x)^{3/2} \left (\frac{3 b^2 c}{a}+\frac{5 a d^2}{c}+40 b d\right )}{24 x}+\sqrt{b} d^{3/2} (3 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{b} \sqrt{c+d x}}\right )-\frac{(a+b x)^{3/2} (c+d x)^{5/2}}{3 x^3}-\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+3 b c)}{12 c x^2} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^4,x]

[Out]

(d*(b^2*c^2 + 26*a*b*c*d + 5*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(8*a*c) - (((
3*b^2*c)/a + 40*b*d + (5*a*d^2)/c)*Sqrt[a + b*x]*(c + d*x)^(3/2))/(24*x) - ((3*b
*c + 5*a*d)*Sqrt[a + b*x]*(c + d*x)^(5/2))/(12*c*x^2) - ((a + b*x)^(3/2)*(c + d*
x)^(5/2))/(3*x^3) + ((b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*Arc
Tanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*a^(3/2)*Sqrt[c]) + Sqr
t[b]*d^(3/2)*(5*b*c + 3*a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d
*x])]

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Rubi in Sympy [A]  time = 165.765, size = 280, normalized size = 0.96 \[ \sqrt{b} d^{\frac{3}{2}} \left (3 a d + 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{b} \sqrt{c + d x}} \right )} - \frac{\left (a + b x\right )^{\frac{3}{2}} \left (c + d x\right )^{\frac{5}{2}}}{3 x^{3}} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{5}{2}} \left (5 a d + 3 b c\right )}{12 c x^{2}} + \frac{d \sqrt{a + b x} \sqrt{c + d x} \left (5 a^{2} d^{2} + 26 a b c d + b^{2} c^{2}\right )}{8 a c} - \frac{\sqrt{a + b x} \left (c + d x\right )^{\frac{3}{2}} \left (5 a^{2} d^{2} + 40 a b c d + 3 b^{2} c^{2}\right )}{24 a c x} - \frac{\left (5 a^{3} d^{3} + 45 a^{2} b c d^{2} + 15 a b^{2} c^{2} d - b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + b x}}{\sqrt{a} \sqrt{c + d x}} \right )}}{8 a^{\frac{3}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(d*x+c)**(5/2)/x**4,x)

[Out]

sqrt(b)*d**(3/2)*(3*a*d + 5*b*c)*atanh(sqrt(d)*sqrt(a + b*x)/(sqrt(b)*sqrt(c + d
*x))) - (a + b*x)**(3/2)*(c + d*x)**(5/2)/(3*x**3) - sqrt(a + b*x)*(c + d*x)**(5
/2)*(5*a*d + 3*b*c)/(12*c*x**2) + d*sqrt(a + b*x)*sqrt(c + d*x)*(5*a**2*d**2 + 2
6*a*b*c*d + b**2*c**2)/(8*a*c) - sqrt(a + b*x)*(c + d*x)**(3/2)*(5*a**2*d**2 + 4
0*a*b*c*d + 3*b**2*c**2)/(24*a*c*x) - (5*a**3*d**3 + 45*a**2*b*c*d**2 + 15*a*b**
2*c**2*d - b**3*c**3)*atanh(sqrt(c)*sqrt(a + b*x)/(sqrt(a)*sqrt(c + d*x)))/(8*a*
*(3/2)*sqrt(c))

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Mathematica [A]  time = 0.26036, size = 297, normalized size = 1.02 \[ \frac{1}{16} \left (-\frac{2 \sqrt{a+b x} \sqrt{c+d x} \left (a^2 \left (8 c^2+26 c d x+33 d^2 x^2\right )+2 a b x \left (7 c^2+34 c d x-12 d^2 x^2\right )+3 b^2 c^2 x^2\right )}{3 a x^3}+\frac{\log (x) \left (5 a^3 d^3+45 a^2 b c d^2+15 a b^2 c^2 d-b^3 c^3\right )}{a^{3/2} \sqrt{c}}+\frac{\left (-5 a^3 d^3-45 a^2 b c d^2-15 a b^2 c^2 d+b^3 c^3\right ) \log \left (2 \sqrt{a} \sqrt{c} \sqrt{a+b x} \sqrt{c+d x}+2 a c+a d x+b c x\right )}{a^{3/2} \sqrt{c}}+8 \sqrt{b} d^{3/2} (3 a d+5 b c) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x} \sqrt{c+d x}+a d+b c+2 b d x\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(c + d*x)^(5/2))/x^4,x]

[Out]

((-2*Sqrt[a + b*x]*Sqrt[c + d*x]*(3*b^2*c^2*x^2 + 2*a*b*x*(7*c^2 + 34*c*d*x - 12
*d^2*x^2) + a^2*(8*c^2 + 26*c*d*x + 33*d^2*x^2)))/(3*a*x^3) + ((-(b^3*c^3) + 15*
a*b^2*c^2*d + 45*a^2*b*c*d^2 + 5*a^3*d^3)*Log[x])/(a^(3/2)*Sqrt[c]) + ((b^3*c^3
- 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*a^3*d^3)*Log[2*a*c + b*c*x + a*d*x + 2*Sqr
t[a]*Sqrt[c]*Sqrt[a + b*x]*Sqrt[c + d*x]])/(a^(3/2)*Sqrt[c]) + 8*Sqrt[b]*d^(3/2)
*(5*b*c + 3*a*d)*Log[b*c + a*d + 2*b*d*x + 2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*x]*Sqrt[
c + d*x]])/16

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Maple [B]  time = 0.024, size = 706, normalized size = 2.4 \[ -{\frac{1}{48\,a{x}^{3}}\sqrt{bx+a}\sqrt{dx+c} \left ( 15\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{3}{d}^{3}\sqrt{bd}+135\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{a}^{2}bc{d}^{2}\sqrt{bd}+45\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}a{b}^{2}{c}^{2}d\sqrt{bd}-3\,\ln \left ({\frac{adx+bcx+2\,\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+2\,ac}{x}} \right ){x}^{3}{b}^{3}{c}^{3}\sqrt{bd}-72\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{3}{a}^{2}b{d}^{3}\sqrt{ac}-120\,\ln \left ( 1/2\,{\frac{2\,bdx+2\,\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}+ad+bc}{\sqrt{bd}}} \right ){x}^{3}a{b}^{2}c{d}^{2}\sqrt{ac}-48\,{x}^{3}ab{d}^{2}\sqrt{bd}\sqrt{ac}\sqrt{d{x}^{2}b+adx+bcx+ac}+66\,\sqrt{d{x}^{2}b+adx+bcx+ac}{d}^{2}\sqrt{bd}{a}^{2}\sqrt{ac}{x}^{2}+136\,\sqrt{d{x}^{2}b+adx+bcx+ac}db\sqrt{bd}a\sqrt{ac}{x}^{2}c+6\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}{b}^{2}\sqrt{bd}\sqrt{ac}{x}^{2}+52\,\sqrt{d{x}^{2}b+adx+bcx+ac}d\sqrt{bd}{a}^{2}\sqrt{ac}xc+28\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}b\sqrt{bd}a\sqrt{ac}x+16\,{c}^{2}\sqrt{d{x}^{2}b+adx+bcx+ac}\sqrt{bd}{a}^{2}\sqrt{ac} \right ){\frac{1}{\sqrt{d{x}^{2}b+adx+bcx+ac}}}{\frac{1}{\sqrt{bd}}}{\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^4,x)

[Out]

-1/48*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a*(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^3*d^3*(b*d)^(1/2)+135*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)-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*b^3*c^3*(b*d)^(1/2)-72*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*a^2*b*d^3*(a*c)^(1/2)-120*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*a*b^2*c*d^2*(a*c)^(1/2)-48*x^3*a*b*d^2*(b*d)^(1/2)*(a*c)^(1/2)*(b*d*x^2+a*
d*x+b*c*x+a*c)^(1/2)+66*(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+136*(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+6*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+52*(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+28*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)/x^3/(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)^(3/2)*(d*x + c)^(5/2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.57168, 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^4,x, algorithm="fricas")

[Out]

[1/96*(24*(5*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sqrt(b*d)*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*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*
c*d^2 - 5*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*(24*a*b*d^2*x^3 - 8*a^2*c^2 - (3*b^2*c^2 + 68*a
*b*c*d + 33*a^2*d^2)*x^2 - 2*(7*a*b*c^2 + 13*a^2*c*d)*x)*sqrt(a*c)*sqrt(b*x + a)
*sqrt(d*x + c))/(sqrt(a*c)*a*x^3), 1/96*(48*(5*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*sq
rt(-b*d)*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*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*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*(24*a*b*d^2*x^3 - 8*a^2*c^2 - (3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)*x^2 - 2*(7
*a*b*c^2 + 13*a^2*c*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(a*c)*a*x^
3), 1/48*(12*(5*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*sqrt(b*d)*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*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^
2*b*c*d^2 - 5*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*(24*a*b*d^2*x^3 - 8*a^2*c^2 - (3*b^2*c^2 + 68*a*
b*c*d + 33*a^2*d^2)*x^2 - 2*(7*a*b*c^2 + 13*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)
*sqrt(d*x + c))/(sqrt(-a*c)*a*x^3), 1/48*(24*(5*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*
sqrt(-b*d)*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*(b^3*c^3 - 15*a*b^2*c^2*d - 45*a^2*b*c*d^2 - 5*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*(
24*a*b*d^2*x^3 - 8*a^2*c^2 - (3*b^2*c^2 + 68*a*b*c*d + 33*a^2*d^2)*x^2 - 2*(7*a*
b*c^2 + 13*a^2*c*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d*x + c))/(sqrt(-a*c)*a*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)**(3/2)*(d*x+c)**(5/2)/x**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.75236, 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^4,x, algorithm="giac")

[Out]

sage0*x