3.2198 \(\int \frac{(a+b x)^{3/2} (A+B x)}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{3 (b d-a e) (-a B e-4 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 \sqrt{b} e^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac{2 (a+b x)^{5/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) - (3*(5*b*B*d - 4
*A*b*e - a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*e^3) + ((5*b*B*d - 4*A*b*e - a*B
*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e^2*(b*d - a*e)) + (3*(b*d - a*e)*(5*b*B*d
- 4*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4
*Sqrt[b]*e^(7/2))

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Rubi [A]  time = 0.414398, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b d-a e) (-a B e-4 A b e+5 b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{4 \sqrt{b} e^{7/2}}-\frac{3 \sqrt{a+b x} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{4 e^3}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (-a B e-4 A b e+5 b B d)}{2 e^2 (b d-a e)}-\frac{2 (a+b x)^{5/2} (B d-A e)}{e \sqrt{d+e x} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

(-2*(B*d - A*e)*(a + b*x)^(5/2))/(e*(b*d - a*e)*Sqrt[d + e*x]) - (3*(5*b*B*d - 4
*A*b*e - a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(4*e^3) + ((5*b*B*d - 4*A*b*e - a*B
*e)*(a + b*x)^(3/2)*Sqrt[d + e*x])/(2*e^2*(b*d - a*e)) + (3*(b*d - a*e)*(5*b*B*d
- 4*A*b*e - a*B*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(4
*Sqrt[b]*e^(7/2))

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Rubi in Sympy [A]  time = 37.5317, size = 190, normalized size = 0.94 \[ - \frac{2 \left (a + b x\right )^{\frac{5}{2}} \left (A e - B d\right )}{e \sqrt{d + e x} \left (a e - b d\right )} + \frac{\left (a + b x\right )^{\frac{3}{2}} \sqrt{d + e x} \left (4 A b e + B a e - 5 B b d\right )}{2 e^{2} \left (a e - b d\right )} + \frac{3 \sqrt{a + b x} \sqrt{d + e x} \left (4 A b e + B a e - 5 B b d\right )}{4 e^{3}} + \frac{3 \left (a e - b d\right ) \left (4 A b e + B a e - 5 B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a + b x}}{\sqrt{b} \sqrt{d + e x}} \right )}}{4 \sqrt{b} e^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(3/2),x)

[Out]

-2*(a + b*x)**(5/2)*(A*e - B*d)/(e*sqrt(d + e*x)*(a*e - b*d)) + (a + b*x)**(3/2)
*sqrt(d + e*x)*(4*A*b*e + B*a*e - 5*B*b*d)/(2*e**2*(a*e - b*d)) + 3*sqrt(a + b*x
)*sqrt(d + e*x)*(4*A*b*e + B*a*e - 5*B*b*d)/(4*e**3) + 3*(a*e - b*d)*(4*A*b*e +
B*a*e - 5*B*b*d)*atanh(sqrt(e)*sqrt(a + b*x)/(sqrt(b)*sqrt(d + e*x)))/(4*sqrt(b)
*e**(7/2))

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Mathematica [A]  time = 0.260112, size = 158, normalized size = 0.78 \[ \frac{\sqrt{a+b x} \left (a e (-8 A e+13 B d+5 B e x)+4 A b e (3 d+e x)+b B \left (-15 d^2-5 d e x+2 e^2 x^2\right )\right )}{4 e^3 \sqrt{d+e x}}+\frac{3 (a e-b d) (a B e+4 A b e-5 b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{8 \sqrt{b} e^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^(3/2)*(A + B*x))/(d + e*x)^(3/2),x]

[Out]

(Sqrt[a + b*x]*(4*A*b*e*(3*d + e*x) + a*e*(13*B*d - 8*A*e + 5*B*e*x) + b*B*(-15*
d^2 - 5*d*e*x + 2*e^2*x^2)))/(4*e^3*Sqrt[d + e*x]) + (3*(-(b*d) + a*e)*(-5*b*B*d
+ 4*A*b*e + a*B*e)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sqrt[a + b*x]*Sq
rt[d + e*x]])/(8*Sqrt[b]*e^(7/2))

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Maple [B]  time = 0.033, size = 740, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)*(B*x+A)/(e*x+d)^(3/2),x)

[Out]

1/8*(b*x+a)^(1/2)*(12*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*
e+b*d)/(b*e)^(1/2))*x*a*b*e^3-12*A*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*
e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^2*d*e^2+3*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a^2*e^3-18*B*ln(1/2*(2*b*x*e+2*((b*
x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*a*b*d*e^2+15*B*ln(1/2*(2
*b*x*e+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*x*b^2*d^2*e+4
*B*x^2*b*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+12*A*ln(1/2*(2*b*x*e+2*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d*e^2-12*A*ln(1/2*(2*b*x*e
+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^2*e+8*A*x*b*e
^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+3*B*ln(1/2*(2*b*x*e+2*((b*x+a)*(e*x+d))^(
1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*d*e^2-18*B*ln(1/2*(2*b*x*e+2*((b*x+a)
*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*d^2*e+15*B*ln(1/2*(2*b*x*e
+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d^3+10*B*x*a*e^
2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-10*B*x*b*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)-16*A*a*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+24*A*b*d*e*((b*x+a)*(e*x+d
))^(1/2)*(b*e)^(1/2)+26*B*a*d*e*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)-30*B*b*d^2*(
(b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2))/((b*x+a)*(e*x+d))^(1/2)/(b*e)^(1/2)/(e*x+d)^
(1/2)/e^3

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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)*(b*x + a)^(3/2)/(e*x + d)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.756915, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (2 \, B b e^{2} x^{2} - 15 \, B b d^{2} - 8 \, A a e^{2} +{\left (13 \, B a + 12 \, A b\right )} d e -{\left (5 \, B b d e -{\left (5 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (5 \, B b^{2} d^{3} - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e +{\left (B a^{2} + 4 \, A a b\right )} d e^{2} +{\left (5 \, B b^{2} d^{2} e - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} +{\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{16 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{b e}}, \frac{2 \,{\left (2 \, B b e^{2} x^{2} - 15 \, B b d^{2} - 8 \, A a e^{2} +{\left (13 \, B a + 12 \, A b\right )} d e -{\left (5 \, B b d e -{\left (5 \, B a + 4 \, A b\right )} e^{2}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (5 \, B b^{2} d^{3} - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d^{2} e +{\left (B a^{2} + 4 \, A a b\right )} d e^{2} +{\left (5 \, B b^{2} d^{2} e - 2 \,{\left (3 \, B a b + 2 \, A b^{2}\right )} d e^{2} +{\left (B a^{2} + 4 \, A a b\right )} e^{3}\right )} x\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{8 \,{\left (e^{4} x + d e^{3}\right )} \sqrt{-b e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)/(e*x + d)^(3/2),x, algorithm="fricas")

[Out]

[1/16*(4*(2*B*b*e^2*x^2 - 15*B*b*d^2 - 8*A*a*e^2 + (13*B*a + 12*A*b)*d*e - (5*B*
b*d*e - (5*B*a + 4*A*b)*e^2)*x)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(5*B*b
^2*d^3 - 2*(3*B*a*b + 2*A*b^2)*d^2*e + (B*a^2 + 4*A*a*b)*d*e^2 + (5*B*b^2*d^2*e
- 2*(3*B*a*b + 2*A*b^2)*d*e^2 + (B*a^2 + 4*A*a*b)*e^3)*x)*log(4*(2*b^2*e^2*x + b
^2*d*e + a*b*e^2)*sqrt(b*x + a)*sqrt(e*x + d) + (8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b
*d*e + a^2*e^2 + 8*(b^2*d*e + a*b*e^2)*x)*sqrt(b*e)))/((e^4*x + d*e^3)*sqrt(b*e)
), 1/8*(2*(2*B*b*e^2*x^2 - 15*B*b*d^2 - 8*A*a*e^2 + (13*B*a + 12*A*b)*d*e - (5*B
*b*d*e - (5*B*a + 4*A*b)*e^2)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 3*(5*B
*b^2*d^3 - 2*(3*B*a*b + 2*A*b^2)*d^2*e + (B*a^2 + 4*A*a*b)*d*e^2 + (5*B*b^2*d^2*
e - 2*(3*B*a*b + 2*A*b^2)*d*e^2 + (B*a^2 + 4*A*a*b)*e^3)*x)*arctan(1/2*(2*b*e*x
+ b*d + a*e)*sqrt(-b*e)/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/((e^4*x + d*e^3)*sqr
t(-b*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \left (a + b x\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**(3/2)*(B*x+A)/(e*x+d)**(3/2),x)

[Out]

Integral((A + B*x)*(a + b*x)**(3/2)/(d + e*x)**(3/2), x)

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GIAC/XCAS [A]  time = 0.253373, size = 367, normalized size = 1.82 \[ \frac{{\left ({\left (\frac{2 \,{\left (b x + a\right )} B b{\left | b \right |} e^{4}}{b^{8} d e^{6} - a b^{7} e^{7}} - \frac{5 \, B b^{2} d{\left | b \right |} e^{3} - B a b{\left | b \right |} e^{4} - 4 \, A b^{2}{\left | b \right |} e^{4}}{b^{8} d e^{6} - a b^{7} e^{7}}\right )}{\left (b x + a\right )} - \frac{3 \,{\left (5 \, B b^{3} d^{2}{\left | b \right |} e^{2} - 6 \, B a b^{2} d{\left | b \right |} e^{3} - 4 \, A b^{3} d{\left | b \right |} e^{3} + B a^{2} b{\left | b \right |} e^{4} + 4 \, A a b^{2}{\left | b \right |} e^{4}\right )}}{b^{8} d e^{6} - a b^{7} e^{7}}\right )} \sqrt{b x + a}}{1536 \, \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e}} - \frac{{\left (5 \, B b d{\left | b \right |} - B a{\left | b \right |} e - 4 \, A b{\left | b \right |} e\right )} e^{\left (-\frac{9}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{512 \, b^{\frac{13}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1536*((2*(b*x + a)*B*b*abs(b)*e^4/(b^8*d*e^6 - a*b^7*e^7) - (5*B*b^2*d*abs(b)*
e^3 - B*a*b*abs(b)*e^4 - 4*A*b^2*abs(b)*e^4)/(b^8*d*e^6 - a*b^7*e^7))*(b*x + a)
- 3*(5*B*b^3*d^2*abs(b)*e^2 - 6*B*a*b^2*d*abs(b)*e^3 - 4*A*b^3*d*abs(b)*e^3 + B*
a^2*b*abs(b)*e^4 + 4*A*a*b^2*abs(b)*e^4)/(b^8*d*e^6 - a*b^7*e^7))*sqrt(b*x + a)/
sqrt(b^2*d + (b*x + a)*b*e - a*b*e) - 1/512*(5*B*b*d*abs(b) - B*a*abs(b)*e - 4*A
*b*abs(b)*e)*e^(-9/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x
+ a)*b*e - a*b*e)))/b^(13/2)