3.2220 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=193 \[ -\frac{(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e} \]

[Out]

-((b*d - a*e)*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^3*e)
- ((b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^2*e) + (B*S
qrt[a + b*x]*(d + e*x)^(5/2))/(3*b*e) - ((b*d - a*e)^2*(b*B*d - 6*A*b*e + 5*a*B*
e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(7/2)*e^(3/2))

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Rubi [A]  time = 0.393761, antiderivative size = 193, 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 \[ -\frac{(b d-a e)^2 (5 a B e-6 A b e+b B d) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{7/2} e^{3/2}}-\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (5 a B e-6 A b e+b B d)}{8 b^3 e}-\frac{\sqrt{a+b x} (d+e x)^{3/2} (5 a B e-6 A b e+b B d)}{12 b^2 e}+\frac{B \sqrt{a+b x} (d+e x)^{5/2}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

-((b*d - a*e)*(b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^3*e)
- ((b*B*d - 6*A*b*e + 5*a*B*e)*Sqrt[a + b*x]*(d + e*x)^(3/2))/(12*b^2*e) + (B*S
qrt[a + b*x]*(d + e*x)^(5/2))/(3*b*e) - ((b*d - a*e)^2*(b*B*d - 6*A*b*e + 5*a*B*
e)*ArcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(7/2)*e^(3/2))

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Rubi in Sympy [A]  time = 28.9301, size = 182, normalized size = 0.94 \[ \frac{B \sqrt{a + b x} \left (d + e x\right )^{\frac{5}{2}}}{3 b e} + \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (6 A b e - 5 B a e - B b d\right )}{12 b^{2} e} - \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (6 A b e - 5 B a e - B b d\right )}{8 b^{3} e} + \frac{\left (a e - b d\right )^{2} \left (6 A b e - 5 B a e - B b d\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{8 b^{\frac{7}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**(1/2),x)

[Out]

B*sqrt(a + b*x)*(d + e*x)**(5/2)/(3*b*e) + sqrt(a + b*x)*(d + e*x)**(3/2)*(6*A*b
*e - 5*B*a*e - B*b*d)/(12*b**2*e) - sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*d)*(6*A
*b*e - 5*B*a*e - B*b*d)/(8*b**3*e) + (a*e - b*d)**2*(6*A*b*e - 5*B*a*e - B*b*d)*
atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(8*b**(7/2)*e**(3/2))

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Mathematica [A]  time = 0.266014, size = 178, normalized size = 0.92 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (15 a^2 B e^2-2 a b e (9 A e+11 B d+5 B e x)+b^2 \left (6 A e (5 d+2 e x)+B \left (3 d^2+14 d e x+8 e^2 x^2\right )\right )\right )}{24 b^3 e}-\frac{(b d-a e)^2 (5 a B e-6 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{7/2} e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(15*a^2*B*e^2 - 2*a*b*e*(11*B*d + 9*A*e + 5*B*e*x)
+ b^2*(6*A*e*(5*d + 2*e*x) + B*(3*d^2 + 14*d*e*x + 8*e^2*x^2))))/(24*b^3*e) - ((
b*d - a*e)^2*(b*B*d - 6*A*b*e + 5*a*B*e)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqr
t[e]*Sqrt[a + b*x]*Sqrt[d + e*x]])/(16*b^(7/2)*e^(3/2))

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Maple [B]  time = 0.031, size = 636, normalized size = 3.3 \[{\frac{1}{48\,{b}^{3}e}\sqrt{ex+d}\sqrt{bx+a} \left ( 16\,B{x}^{2}{b}^{2}{e}^{2}\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+18\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}A{e}^{3}b-36\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aA{b}^{2}d{e}^{2}+18\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{2}Ae+24\,A\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }x{b}^{2}{e}^{2}\sqrt{be}-15\,B{e}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}+27\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}Bd{e}^{2}b-9\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) aB{b}^{2}{d}^{2}e-3\,\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){b}^{3}{d}^{3}B-20\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xab{e}^{2}\sqrt{be}+28\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }xd{b}^{2}e\sqrt{be}-36\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Aa{e}^{2}\sqrt{be}b+60\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }A{b}^{2}de\sqrt{be}+30\,B\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }{a}^{2}{e}^{2}\sqrt{be}-44\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }Bade\sqrt{be}b+6\,\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }B{b}^{2}{d}^{2}\sqrt{be} \right ){\frac{1}{\sqrt{ \left ( bx+a \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^(3/2)/(b*x+a)^(1/2),x)

[Out]

1/48*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(16*B*x^2*b^2*e^2*((b*x+a)*(e*x+d))^(1/2)*(b*e)
^(1/2)+18*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*A*e^3*b-36*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*A*b^2*d*e^2+18*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^3*d^2*A*e+24*A*((b*x+a)*(e*x+d))^(1/2)*x*b^2*e^2
*(b*e)^(1/2)-15*B*e^3*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^3+27*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*B*d*e^2*b-9*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*b^2*d^2*e-3*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^3*d^3*B-20*B*((b*x+a)*(e*x+d))^(1
/2)*x*a*b*e^2*(b*e)^(1/2)+28*B*((b*x+a)*(e*x+d))^(1/2)*x*d*b^2*e*(b*e)^(1/2)-36*
((b*x+a)*(e*x+d))^(1/2)*A*a*e^2*(b*e)^(1/2)*b+60*((b*x+a)*(e*x+d))^(1/2)*A*b^2*d
*e*(b*e)^(1/2)+30*B*((b*x+a)*(e*x+d))^(1/2)*a^2*e^2*(b*e)^(1/2)-44*((b*x+a)*(e*x
+d))^(1/2)*B*a*d*e*(b*e)^(1/2)*b+6*((b*x+a)*(e*x+d))^(1/2)*B*b^2*d^2*(b*e)^(1/2)
)/((b*x+a)*(e*x+d))^(1/2)/b^3/(b*e)^(1/2)/e

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.520614, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 2 \,{\left (11 \, B a b - 15 \, A b^{2}\right )} d e + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} e^{2} + 2 \,{\left (7 \, B b^{2} d e -{\left (5 \, B a b - 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} - 3 \,{\left (B b^{3} d^{3} + 3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \log \left (4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{96 \, \sqrt{b e} b^{3} e}, \frac{2 \,{\left (8 \, B b^{2} e^{2} x^{2} + 3 \, B b^{2} d^{2} - 2 \,{\left (11 \, B a b - 15 \, A b^{2}\right )} d e + 3 \,{\left (5 \, B a^{2} - 6 \, A a b\right )} e^{2} + 2 \,{\left (7 \, B b^{2} d e -{\left (5 \, B a b - 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} - 3 \,{\left (B b^{3} d^{3} + 3 \,{\left (B a b^{2} - 2 \, A b^{3}\right )} d^{2} e - 3 \,{\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} e^{3}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{48 \, \sqrt{-b e} b^{3} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*B*b^2*e^2*x^2 + 3*B*b^2*d^2 - 2*(11*B*a*b - 15*A*b^2)*d*e + 3*(5*B*a
^2 - 6*A*a*b)*e^2 + 2*(7*B*b^2*d*e - (5*B*a*b - 6*A*b^2)*e^2)*x)*sqrt(b*e)*sqrt(
b*x + a)*sqrt(e*x + d) - 3*(B*b^3*d^3 + 3*(B*a*b^2 - 2*A*b^3)*d^2*e - 3*(3*B*a^2
*b - 4*A*a*b^2)*d*e^2 + (5*B*a^3 - 6*A*a^2*b)*e^3)*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)))/(sqrt(b*e)*b^3*e), 1/48*(2*(8*B*b
^2*e^2*x^2 + 3*B*b^2*d^2 - 2*(11*B*a*b - 15*A*b^2)*d*e + 3*(5*B*a^2 - 6*A*a*b)*e
^2 + 2*(7*B*b^2*d*e - (5*B*a*b - 6*A*b^2)*e^2)*x)*sqrt(-b*e)*sqrt(b*x + a)*sqrt(
e*x + d) - 3*(B*b^3*d^3 + 3*(B*a*b^2 - 2*A*b^3)*d^2*e - 3*(3*B*a^2*b - 4*A*a*b^2
)*d*e^2 + (5*B*a^3 - 6*A*a^2*b)*e^3)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)
/(sqrt(b*x + a)*sqrt(e*x + d)*b*e)))/(sqrt(-b*e)*b^3*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**(3/2)/(b*x+a)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.296499, size = 790, normalized size = 4.09 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(48*((b^2*d - a*b*e)*e^(-1/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt
(b^2*d + (b*x + a)*b*e - a*b*e)))/sqrt(b) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*
sqrt(b*x + a))*A*d*abs(b)/b^2 - 2*(sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x
+ a)*(2*(b*x + a)*(4*(b*x + a)/b^2 + (b^6*d*e^3 - 13*a*b^5*e^4)*e^(-4)/b^7) - 3*
(b^7*d^2*e^2 + 2*a*b^6*d*e^3 - 11*a^2*b^5*e^4)*e^(-4)/b^7) - 3*(b^3*d^3 + a*b^2*
d^2*e + 3*a^2*b*d*e^2 - 5*a^3*e^3)*e^(-5/2)*ln(abs(-sqrt(b*x + a)*sqrt(b)*e^(1/2
) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(3/2))*B*abs(b)*e/b^2 - (sqrt(b^2*d
+ (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 + (b*d*e - 5*a*e^
2)*e^(-4)/b^4) + (b^2*d^2 + 2*a*b*d*e - 3*a^2*e^2)*e^(-7/2)*ln(abs(-sqrt(b*x + a
)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2))*B*d*abs(b)/b^
3 - (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sqrt(b*x + a)*(2*(b*x + a)*e^(-2)/b^4 +
(b*d*e - 5*a*e^2)*e^(-4)/b^4) + (b^2*d^2 + 2*a*b*d*e - 3*a^2*e^2)*e^(-7/2)*ln(a
bs(-sqrt(b*x + a)*sqrt(b)*e^(1/2) + sqrt(b^2*d + (b*x + a)*b*e - a*b*e)))/b^(7/2
))*A*abs(b)*e/b^3)/b