3.2185 \(\int \sqrt{a+b x} (A+B x) \sqrt{d+e x} \, dx\)

Optimal. Leaf size=196 \[ -\frac{(b d-a e)^2 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{5/2} e^{5/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{8 b^2 e^2}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (2 A b e-B (a e+b d))}{4 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e} \]

[Out]

((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^2*e^2)
+ ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(4*b^2*e) + (B*(a +
b*x)^(3/2)*(d + e*x)^(3/2))/(3*b*e) - ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*A
rcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(5/2)*e^(5/2))

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Rubi [A]  time = 0.393722, antiderivative size = 196, 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 \[ -\frac{(b d-a e)^2 (2 A b e-B (a e+b d)) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a+b x}}{\sqrt{b} \sqrt{d+e x}}\right )}{8 b^{5/2} e^{5/2}}+\frac{\sqrt{a+b x} \sqrt{d+e x} (b d-a e) (2 A b e-B (a e+b d))}{8 b^2 e^2}+\frac{(a+b x)^{3/2} \sqrt{d+e x} (2 A b e-B (a e+b d))}{4 b^2 e}+\frac{B (a+b x)^{3/2} (d+e x)^{3/2}}{3 b e} \]

Antiderivative was successfully verified.

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

[Out]

((b*d - a*e)*(2*A*b*e - B*(b*d + a*e))*Sqrt[a + b*x]*Sqrt[d + e*x])/(8*b^2*e^2)
+ ((2*A*b*e - B*(b*d + a*e))*(a + b*x)^(3/2)*Sqrt[d + e*x])/(4*b^2*e) + (B*(a +
b*x)^(3/2)*(d + e*x)^(3/2))/(3*b*e) - ((b*d - a*e)^2*(2*A*b*e - B*(b*d + a*e))*A
rcTanh[(Sqrt[e]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[d + e*x])])/(8*b^(5/2)*e^(5/2))

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Rubi in Sympy [A]  time = 29.3348, size = 175, normalized size = 0.89 \[ \frac{B \left (a + b x\right )^{\frac{3}{2}} \left (d + e x\right )^{\frac{3}{2}}}{3 b e} - \frac{\sqrt{a + b x} \left (d + e x\right )^{\frac{3}{2}} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right )}{2 b e^{2}} + \frac{\sqrt{a + b x} \sqrt{d + e x} \left (a e - b d\right ) \left (2 A b e - B a e - B b d\right )}{8 b^{2} e^{2}} + \frac{\left (a e - b d\right )^{2} \left (- A b e + \frac{B \left (a e + b d\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{e} \sqrt{a + b x}} \right )}}{4 b^{\frac{5}{2}} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x)**(3/2)*(d + e*x)**(3/2)/(3*b*e) - sqrt(a + b*x)*(d + e*x)**(3/2)*(-A
*b*e + B*(a*e + b*d)/2)/(2*b*e**2) + sqrt(a + b*x)*sqrt(d + e*x)*(a*e - b*d)*(2*
A*b*e - B*a*e - B*b*d)/(8*b**2*e**2) + (a*e - b*d)**2*(-A*b*e + B*(a*e + b*d)/2)
*atanh(sqrt(b)*sqrt(d + e*x)/(sqrt(e)*sqrt(a + b*x)))/(4*b**(5/2)*e**(5/2))

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Mathematica [A]  time = 0.197005, size = 173, normalized size = 0.88 \[ \frac{\sqrt{a+b x} \sqrt{d+e x} \left (-3 a^2 B e^2+2 a b e (3 A e+B (d+e x))+b^2 \left (6 A e (d+2 e x)+B \left (-3 d^2+2 d e x+8 e^2 x^2\right )\right )\right )}{24 b^2 e^2}+\frac{(b d-a e)^2 (a B e-2 A b e+b B d) \log \left (2 \sqrt{b} \sqrt{e} \sqrt{a+b x} \sqrt{d+e x}+a e+b d+2 b e x\right )}{16 b^{5/2} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x]*Sqrt[d + e*x]*(-3*a^2*B*e^2 + 2*a*b*e*(3*A*e + B*(d + e*x)) + b^2
*(6*A*e*(d + 2*e*x) + B*(-3*d^2 + 2*d*e*x + 8*e^2*x^2))))/(24*b^2*e^2) + ((b*d -
a*e)^2*(b*B*d - 2*A*b*e + a*B*e)*Log[b*d + a*e + 2*b*e*x + 2*Sqrt[b]*Sqrt[e]*Sq
rt[a + b*x]*Sqrt[d + e*x]])/(16*b^(5/2)*e^(5/2))

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Maple [B]  time = 0.019, size = 755, normalized size = 3.9 \[ -{\frac{1}{48\,{b}^{2}{e}^{2}}\sqrt{bx+a}\sqrt{ex+d} \left ( -16\,B{x}^{2}{b}^{2}{e}^{2}\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+6\,A{e}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}b-12\,A\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) ad{b}^{2}{e}^{2}+6\,A{b}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){d}^{2}e-24\,A\sqrt{be{x}^{2}+aex+bdx+ad}x{b}^{2}{e}^{2}\sqrt{be}-3\,B{e}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{3}+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){a}^{2}db{e}^{2}+3\,B\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ) a{d}^{2}{b}^{2}e-3\,B{b}^{3}\ln \left ( 1/2\,{\frac{2\,bxe+2\,\sqrt{be{x}^{2}+aex+bdx+ad}\sqrt{be}+ae+bd}{\sqrt{be}}} \right ){d}^{3}-4\,B\sqrt{be{x}^{2}+aex+bdx+ad}xab{e}^{2}\sqrt{be}-4\,B\sqrt{be{x}^{2}+aex+bdx+ad}xd{b}^{2}e\sqrt{be}-12\,A\sqrt{be{x}^{2}+aex+bdx+ad}ab{e}^{2}\sqrt{be}-12\,A\sqrt{be{x}^{2}+aex+bdx+ad}d{b}^{2}e\sqrt{be}+6\,B\sqrt{be{x}^{2}+aex+bdx+ad}{a}^{2}{e}^{2}\sqrt{be}-4\,B\sqrt{be{x}^{2}+aex+bdx+ad}adbe\sqrt{be}+6\,B\sqrt{be{x}^{2}+aex+bdx+ad}{d}^{2}{b}^{2}\sqrt{be} \right ){\frac{1}{\sqrt{be{x}^{2}+aex+bdx+ad}}}{\frac{1}{\sqrt{be}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(b*x+a)^(1/2)*(e*x+d)^(1/2)*(-16*B*x^2*b^2*e^2*(b*e*x^2+a*e*x+b*d*x+a*d)^(
1/2)*(b*e)^(1/2)+6*A*e^3*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)
^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*b-12*A*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a
*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*d*b^2*e^2+6*A*b^3*ln(1/2*(2*b*x*e+
2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*d^2*e-24*A*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*b^2*e^2*(b*e)^(1/2)-3*B*e^3*ln(1/2*(2*b*x*e+2*(
b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^3+3*B*ln(1/2*
(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2
*d*b*e^2+3*B*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*(b*e)^(1/2)+a*e+b
*d)/(b*e)^(1/2))*a*d^2*b^2*e-3*B*b^3*ln(1/2*(2*b*x*e+2*(b*e*x^2+a*e*x+b*d*x+a*d)
^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*d^3-4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)
*x*a*b*e^2*(b*e)^(1/2)-4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*x*d*b^2*e*(b*e)^(1/2)
-12*A*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*b*e^2*(b*e)^(1/2)-12*A*(b*e*x^2+a*e*x+b*
d*x+a*d)^(1/2)*d*b^2*e*(b*e)^(1/2)+6*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a^2*e^2*(
b*e)^(1/2)-4*B*(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)*a*d*b*e*(b*e)^(1/2)+6*B*(b*e*x^2+
a*e*x+b*d*x+a*d)^(1/2)*d^2*b^2*(b*e)^(1/2))/(b*e*x^2+a*e*x+b*d*x+a*d)^(1/2)/b^2/
e^2/(b*e)^(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)*sqrt(b*x + a)*sqrt(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261356, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, B b^{2} e^{2} x^{2} - 3 \, B b^{2} d^{2} + 2 \,{\left (B a b + 3 \, A b^{2}\right )} d e - 3 \,{\left (B a^{2} - 2 \, A a b\right )} e^{2} + 2 \,{\left (B b^{2} d e +{\left (B a b + 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{b e} \sqrt{b x + a} \sqrt{e x + d} - 3 \,{\left (B b^{3} d^{3} -{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e -{\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \log \left (-4 \,{\left (2 \, b^{2} e^{2} x + b^{2} d e + a b e^{2}\right )} \sqrt{b x + a} \sqrt{e x + d} +{\left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} + 8 \,{\left (b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{b e}\right )}{96 \, \sqrt{b e} b^{2} e^{2}}, \frac{2 \,{\left (8 \, B b^{2} e^{2} x^{2} - 3 \, B b^{2} d^{2} + 2 \,{\left (B a b + 3 \, A b^{2}\right )} d e - 3 \,{\left (B a^{2} - 2 \, A a b\right )} e^{2} + 2 \,{\left (B b^{2} d e +{\left (B a b + 6 \, A b^{2}\right )} e^{2}\right )} x\right )} \sqrt{-b e} \sqrt{b x + a} \sqrt{e x + d} + 3 \,{\left (B b^{3} d^{3} -{\left (B a b^{2} + 2 \, A b^{3}\right )} d^{2} e -{\left (B a^{2} b - 4 \, A a b^{2}\right )} d e^{2} +{\left (B a^{3} - 2 \, A a^{2} b\right )} e^{3}\right )} \arctan \left (\frac{{\left (2 \, b e x + b d + a e\right )} \sqrt{-b e}}{2 \, \sqrt{b x + a} \sqrt{e x + d} b e}\right )}{48 \, \sqrt{-b e} b^{2} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(4*(8*B*b^2*e^2*x^2 - 3*B*b^2*d^2 + 2*(B*a*b + 3*A*b^2)*d*e - 3*(B*a^2 - 2
*A*a*b)*e^2 + 2*(B*b^2*d*e + (B*a*b + 6*A*b^2)*e^2)*x)*sqrt(b*e)*sqrt(b*x + a)*s
qrt(e*x + d) - 3*(B*b^3*d^3 - (B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*
d*e^2 + (B*a^3 - 2*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^2*e^2), 1/48*(2*(8*B*b^2*e^2*x^2 - 3*
B*b^2*d^2 + 2*(B*a*b + 3*A*b^2)*d*e - 3*(B*a^2 - 2*A*a*b)*e^2 + 2*(B*b^2*d*e + (
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 -
(B*a*b^2 + 2*A*b^3)*d^2*e - (B*a^2*b - 4*A*a*b^2)*d*e^2 + (B*a^3 - 2*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^2*e^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257982, size = 439, normalized size = 2.24 \[ \frac{\frac{20 \,{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (\frac{2 \,{\left (b x + a\right )} e^{\left (-2\right )}}{b^{4}} + \frac{{\left (b d e - a e^{2}\right )} e^{\left (-4\right )}}{b^{4}}\right )} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-\frac{7}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{7}{2}}}\right )} A{\left | b \right |}}{b^{2}} + \frac{{\left (\sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \sqrt{b x + a}{\left (2 \,{\left (b x + a\right )}{\left (\frac{4 \,{\left (b x + a\right )} e^{\left (-2\right )}}{b^{6}} + \frac{{\left (b d e^{3} - 7 \, a e^{4}\right )} e^{\left (-6\right )}}{b^{6}}\right )} - \frac{3 \,{\left (b^{2} d^{2} e^{2} - a^{2} e^{4}\right )} e^{\left (-6\right )}}{b^{6}}\right )} - \frac{3 \,{\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} e^{\left (-\frac{9}{2}\right )}{\rm ln}\left ({\left | -\sqrt{b x + a} \sqrt{b} e^{\frac{1}{2}} + \sqrt{b^{2} d +{\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac{11}{2}}}\right )} B{\left | b \right |}}{b^{3}}}{1920 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*(20*(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 - a*e^2)*e^(-4)/b^4) + (b^2*d^2 - 2*a*b*d*e + a^2*e^2)*e^(-7/2)*l
n(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))*A*abs(b)/b^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)*e^(-2)/b^6 + (b*d*e^3 - 7*a*e^4)*e^(-6)/b^6) - 3*(b^2*d^2*e^2
- a^2*e^4)*e^(-6)/b^6) - 3*(b^3*d^3 - a*b^2*d^2*e - a^2*b*d*e^2 + a^3*e^3)*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^(11/2))*B*abs(b)/b^3)/b