3.18 \(\int (A+B x) \left (a+b x^2\right )^{5/2} \, dx\)

Optimal. Leaf size=107 \[ \frac{5 a^3 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{16} a^2 A x \sqrt{a+b x^2}+\frac{1}{6} A x \left (a+b x^2\right )^{5/2}+\frac{5}{24} a A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]

[Out]

(5*a^2*A*x*Sqrt[a + b*x^2])/16 + (5*a*A*x*(a + b*x^2)^(3/2))/24 + (A*x*(a + b*x^
2)^(5/2))/6 + (B*(a + b*x^2)^(7/2))/(7*b) + (5*a^3*A*ArcTanh[(Sqrt[b]*x)/Sqrt[a
+ b*x^2]])/(16*Sqrt[b])

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Rubi [A]  time = 0.0975907, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{5 a^3 A \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 \sqrt{b}}+\frac{5}{16} a^2 A x \sqrt{a+b x^2}+\frac{1}{6} A x \left (a+b x^2\right )^{5/2}+\frac{5}{24} a A x \left (a+b x^2\right )^{3/2}+\frac{B \left (a+b x^2\right )^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(a + b*x^2)^(5/2),x]

[Out]

(5*a^2*A*x*Sqrt[a + b*x^2])/16 + (5*a*A*x*(a + b*x^2)^(3/2))/24 + (A*x*(a + b*x^
2)^(5/2))/6 + (B*(a + b*x^2)^(7/2))/(7*b) + (5*a^3*A*ArcTanh[(Sqrt[b]*x)/Sqrt[a
+ b*x^2]])/(16*Sqrt[b])

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Rubi in Sympy [A]  time = 10.5491, size = 100, normalized size = 0.93 \[ \frac{5 A a^{3} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{16 \sqrt{b}} + \frac{5 A a^{2} x \sqrt{a + b x^{2}}}{16} + \frac{5 A a x \left (a + b x^{2}\right )^{\frac{3}{2}}}{24} + \frac{A x \left (a + b x^{2}\right )^{\frac{5}{2}}}{6} + \frac{B \left (a + b x^{2}\right )^{\frac{7}{2}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b*x**2+a)**(5/2),x)

[Out]

5*A*a**3*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(16*sqrt(b)) + 5*A*a**2*x*sqrt(a + b*
x**2)/16 + 5*A*a*x*(a + b*x**2)**(3/2)/24 + A*x*(a + b*x**2)**(5/2)/6 + B*(a + b
*x**2)**(7/2)/(7*b)

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Mathematica [A]  time = 0.136852, size = 108, normalized size = 1.01 \[ \frac{105 a^3 A \sqrt{b} \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )+\sqrt{a+b x^2} \left (48 a^3 B+3 a^2 b x (77 A+48 B x)+2 a b^2 x^3 (91 A+72 B x)+8 b^3 x^5 (7 A+6 B x)\right )}{336 b} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(a + b*x^2)^(5/2),x]

[Out]

(Sqrt[a + b*x^2]*(48*a^3*B + 8*b^3*x^5*(7*A + 6*B*x) + 3*a^2*b*x*(77*A + 48*B*x)
+ 2*a*b^2*x^3*(91*A + 72*B*x)) + 105*a^3*A*Sqrt[b]*Log[b*x + Sqrt[b]*Sqrt[a + b
*x^2]])/(336*b)

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Maple [A]  time = 0.006, size = 85, normalized size = 0.8 \[{\frac{Ax}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,aAx}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}Ax}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,A{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{B}{7\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b*x^2+a)^(5/2),x)

[Out]

1/6*A*x*(b*x^2+a)^(5/2)+5/24*a*A*x*(b*x^2+a)^(3/2)+5/16*a^2*A*x*(b*x^2+a)^(1/2)+
5/16*A*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/7*B*(b*x^2+a)^(7/2)/b

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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^2 + a)^(5/2)*(B*x + A),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265796, size = 1, normalized size = 0.01 \[ \left [\frac{105 \, A a^{3} b \log \left (-2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right ) + 2 \,{\left (48 \, B b^{3} x^{6} + 56 \, A b^{3} x^{5} + 144 \, B a b^{2} x^{4} + 182 \, A a b^{2} x^{3} + 144 \, B a^{2} b x^{2} + 231 \, A a^{2} b x + 48 \, B a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{b}}{672 \, b^{\frac{3}{2}}}, \frac{105 \, A a^{3} b \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) +{\left (48 \, B b^{3} x^{6} + 56 \, A b^{3} x^{5} + 144 \, B a b^{2} x^{4} + 182 \, A a b^{2} x^{3} + 144 \, B a^{2} b x^{2} + 231 \, A a^{2} b x + 48 \, B a^{3}\right )} \sqrt{b x^{2} + a} \sqrt{-b}}{336 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(B*x + A),x, algorithm="fricas")

[Out]

[1/672*(105*A*a^3*b*log(-2*sqrt(b*x^2 + a)*b*x - (2*b*x^2 + a)*sqrt(b)) + 2*(48*
B*b^3*x^6 + 56*A*b^3*x^5 + 144*B*a*b^2*x^4 + 182*A*a*b^2*x^3 + 144*B*a^2*b*x^2 +
231*A*a^2*b*x + 48*B*a^3)*sqrt(b*x^2 + a)*sqrt(b))/b^(3/2), 1/336*(105*A*a^3*b*
arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (48*B*b^3*x^6 + 56*A*b^3*x^5 + 144*B*a*b^2*
x^4 + 182*A*a*b^2*x^3 + 144*B*a^2*b*x^2 + 231*A*a^2*b*x + 48*B*a^3)*sqrt(b*x^2 +
a)*sqrt(-b))/(sqrt(-b)*b)]

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Sympy [A]  time = 18.9186, size = 348, normalized size = 3.25 \[ \frac{A a^{\frac{5}{2}} x \sqrt{1 + \frac{b x^{2}}{a}}}{2} + \frac{3 A a^{\frac{5}{2}} x}{16 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{35 A a^{\frac{3}{2}} b x^{3}}{48 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{17 A \sqrt{a} b^{2} x^{5}}{24 \sqrt{1 + \frac{b x^{2}}{a}}} + \frac{5 A a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{16 \sqrt{b}} + \frac{A b^{3} x^{7}}{6 \sqrt{a} \sqrt{1 + \frac{b x^{2}}{a}}} + B a^{2} \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: b = 0 \\\frac{\left (a + b x^{2}\right )^{\frac{3}{2}}}{3 b} & \text{otherwise} \end{cases}\right ) + 2 B a b \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + B b^{2} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + b x^{2}}}{105 b^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + b x^{2}}}{105 b^{2}} + \frac{a x^{4} \sqrt{a + b x^{2}}}{35 b} + \frac{x^{6} \sqrt{a + b x^{2}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(b*x**2+a)**(5/2),x)

[Out]

A*a**(5/2)*x*sqrt(1 + b*x**2/a)/2 + 3*A*a**(5/2)*x/(16*sqrt(1 + b*x**2/a)) + 35*
A*a**(3/2)*b*x**3/(48*sqrt(1 + b*x**2/a)) + 17*A*sqrt(a)*b**2*x**5/(24*sqrt(1 +
b*x**2/a)) + 5*A*a**3*asinh(sqrt(b)*x/sqrt(a))/(16*sqrt(b)) + A*b**3*x**7/(6*sqr
t(a)*sqrt(1 + b*x**2/a)) + B*a**2*Piecewise((sqrt(a)*x**2/2, Eq(b, 0)), ((a + b*
x**2)**(3/2)/(3*b), True)) + 2*B*a*b*Piecewise((-2*a**2*sqrt(a + b*x**2)/(15*b**
2) + a*x**2*sqrt(a + b*x**2)/(15*b) + x**4*sqrt(a + b*x**2)/5, Ne(b, 0)), (sqrt(
a)*x**4/4, True)) + B*b**2*Piecewise((8*a**3*sqrt(a + b*x**2)/(105*b**3) - 4*a**
2*x**2*sqrt(a + b*x**2)/(105*b**2) + a*x**4*sqrt(a + b*x**2)/(35*b) + x**6*sqrt(
a + b*x**2)/7, Ne(b, 0)), (sqrt(a)*x**6/6, True))

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GIAC/XCAS [A]  time = 0.21724, size = 136, normalized size = 1.27 \[ -\frac{5 \, A a^{3}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, \sqrt{b}} + \frac{1}{336} \,{\left (\frac{48 \, B a^{3}}{b} +{\left (231 \, A a^{2} + 2 \,{\left (72 \, B a^{2} +{\left (91 \, A a b + 4 \,{\left (18 \, B a b +{\left (6 \, B b^{2} x + 7 \, A b^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt{b x^{2} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(B*x + A),x, algorithm="giac")

[Out]

-5/16*A*a^3*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/sqrt(b) + 1/336*(48*B*a^3/b +
(231*A*a^2 + 2*(72*B*a^2 + (91*A*a*b + 4*(18*B*a*b + (6*B*b^2*x + 7*A*b^2)*x)*x)
*x)*x)*x)*sqrt(b*x^2 + a)