3.170 \(\int \sqrt{a+b x^2} \left (c+d x^2\right )^{3/2} \, dx\)

Optimal. Leaf size=336 \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{15 b^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{c^{3/2} \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{15 b} \]

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^2*Sqrt[c + d*x^2])
+ (2*(3*b*c - a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b) + (d*x*(a + b*x^2)
^(3/2)*Sqrt[c + d*x^2])/(5*b) - (Sqrt[c]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqr
t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*Sq
rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(9*b*c -
a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(
15*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.729295, antiderivative size = 336, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{x \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right )}{15 b^2 \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} \left (-2 a^2 d^2+7 a b c d+3 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{c^{3/2} \sqrt{a+b x^2} (9 b c-a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{d x \left (a+b x^2\right )^{3/2} \sqrt{c+d x^2}}{5 b}+\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (3 b c-a d)}{15 b} \]

Antiderivative was successfully verified.

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

[Out]

((3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*x*Sqrt[a + b*x^2])/(15*b^2*Sqrt[c + d*x^2])
+ (2*(3*b*c - a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b) + (d*x*(a + b*x^2)
^(3/2)*Sqrt[c + d*x^2])/(5*b) - (Sqrt[c]*(3*b^2*c^2 + 7*a*b*c*d - 2*a^2*d^2)*Sqr
t[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b^2*Sq
rt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2)*(9*b*c -
a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(
15*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 98.0258, size = 299, normalized size = 0.89 \[ - \frac{a^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - 9 b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 b^{\frac{3}{2}} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{\sqrt{a} \sqrt{c + d x^{2}} \left (2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 b^{\frac{3}{2}} d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{d x \left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}}}{5 b} - \frac{2 x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (\frac{a d}{3} - b c\right )}{5 b} - \frac{x \sqrt{c + d x^{2}} \left (2 a^{2} d^{2} - 7 a b c d - 3 b^{2} c^{2}\right )}{15 b d \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(3/2)*sqrt(c + d*x**2)*(a*d - 9*b*c)*elliptic_f(atan(sqrt(b)*x/sqrt(a)), -a*
d/(b*c) + 1)/(15*b**(3/2)*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)
) + sqrt(a)*sqrt(c + d*x**2)*(2*a**2*d**2 - 7*a*b*c*d - 3*b**2*c**2)*elliptic_e(
atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(15*b**(3/2)*d*sqrt(a*(c + d*x**2)/(c*(
a + b*x**2)))*sqrt(a + b*x**2)) + d*x*(a + b*x**2)**(3/2)*sqrt(c + d*x**2)/(5*b)
- 2*x*sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d/3 - b*c)/(5*b) - x*sqrt(c + d*x**2
)*(2*a**2*d**2 - 7*a*b*c*d - 3*b**2*c**2)/(15*b*d*sqrt(a + b*x**2))

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Mathematica [C]  time = 0.716102, size = 246, normalized size = 0.73 \[ \frac{-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (a^2 d^2+2 a b c d-3 b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (2 a^2 d^2-7 a b c d-3 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d+6 b c+3 b d x^2\right )}{15 b d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*b*c + a*d + 3*b*d*x^2) + I*c*(-3*b^2*c
^2 - 7*a*b*c*d + 2*a^2*d^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*
ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-3*b^2*c^2 + 2*a*b*c*d + a^2*d^2)*Sqrt
[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c
)])/(15*b*Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.022, size = 545, normalized size = 1.6 \[{\frac{1}{15\,d \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) b}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{2}{d}^{3}+4\,\sqrt{-{\frac{b}{a}}}{x}^{5}ab{d}^{3}+9\,\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{2}c{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}{d}^{3}+10\,\sqrt{-{\frac{b}{a}}}{x}^{3}abc{d}^{2}+6\,\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{2}{c}^{2}d+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d-3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}+\sqrt{-{\frac{b}{a}}}x{a}^{2}c{d}^{2}+6\,\sqrt{-{\frac{b}{a}}}xab{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(3*(-b/a)^(1/2)*x^7*b^2*d^3+4*(-b/a)^(1/2)*
x^5*a*b*d^3+9*(-b/a)^(1/2)*x^5*b^2*c*d^2+(-b/a)^(1/2)*x^3*a^2*d^3+10*(-b/a)^(1/2
)*x^3*a*b*c*d^2+6*(-b/a)^(1/2)*x^3*b^2*c^2*d+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(
1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a^2*c*d^2+2*((b*x^2+a)/a)^(1/2)*(
(d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*b*c^2*d-3*((b*x^2
+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c
^3-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^
(1/2))*a^2*c*d^2+7*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(-b/a)^(1
/2),(a*d/b/c)^(1/2))*a*b*c^2*d+3*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti
cE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^3+(-b/a)^(1/2)*x*a^2*c*d^2+6*(-b/a)^(1/
2)*x*a*b*c^2*d)/d/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/b/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x**2)*(c + d*x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2} + a}{\left (d x^{2} + c\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^2 + a)*(d*x^2 + c)^(3/2), x)