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

Optimal. Leaf size=328 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{b}+7 a c-\frac{2 b c^2}{d}\right )}{15 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-3 a d)}{15 d} \]

[Out]

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

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Rubi [A]  time = 0.73919, antiderivative size = 328, 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 \[ \frac{\sqrt{c} \sqrt{a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 b d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{x \sqrt{a+b x^2} \left (\frac{3 a^2 d}{b}+7 a c-\frac{2 b c^2}{d}\right )}{15 \sqrt{c+d x^2}}-\frac{c^{3/2} \sqrt{a+b x^2} (b c-9 a d) F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{15 d^{3/2} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b x \sqrt{a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}-\frac{2 x \sqrt{a+b x^2} \sqrt{c+d x^2} (b c-3 a d)}{15 d} \]

Antiderivative was successfully verified.

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

[Out]

((7*a*c - (2*b*c^2)/d + (3*a^2*d)/b)*x*Sqrt[a + b*x^2])/(15*Sqrt[c + d*x^2]) - (
2*(b*c - 3*a*d)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*d) + (b*x*Sqrt[a + b*x^2]
*(c + d*x^2)^(3/2))/(5*d) + (Sqrt[c]*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a
+ b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*d^(3/2)*
Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*c - 9*a*d)*
Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(15*d^(
3/2)*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 = 91.9174, size = 301, normalized size = 0.92 \[ - \frac{\sqrt{a} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{15 \sqrt{b} d^{2} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{b x \sqrt{a + b x^{2}} \left (c + d x^{2}\right )^{\frac{3}{2}}}{5 d} + \frac{c^{\frac{3}{2}} \sqrt{a + b x^{2}} \left (9 a d - b c\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{15 d^{\frac{3}{2}} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{2 x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - \frac{b c}{3}\right )}{5 d} + \frac{x \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 7 a b c d - 2 b^{2} c^{2}\right )}{15 d^{2} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.798564, size = 243, normalized size = 0.74 \[ \frac{-2 i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2-4 a b c d+b^2 c^2\right ) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} \left (3 a^2 d^2+7 a b c d-2 b^2 c^2\right ) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )+d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (6 a d+b \left (c+3 d x^2\right )\right )}{15 d^2 \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(6*a*d + b*(c + 3*d*x^2)) - I*c*(-2*b^2*c
^2 + 7*a*b*c*d + 3*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)] - (2*I)*c*(b^2*c^2 - 4*a*b*c*d + 3*a^2*d^2)*S
qrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(
b*c)])/(15*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.068, size = 543, normalized size = 1.7 \[{\frac{1}{ \left ( 15\,bd{x}^{4}+15\,ad{x}^{2}+15\,c{x}^{2}b+15\,ac \right ){d}^{2}}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( 3\,\sqrt{-{\frac{b}{a}}}{x}^{7}{b}^{2}{d}^{3}+9\,\sqrt{-{\frac{b}{a}}}{x}^{5}ab{d}^{3}+4\,\sqrt{-{\frac{b}{a}}}{x}^{5}{b}^{2}c{d}^{2}+6\,\sqrt{-{\frac{b}{a}}}{x}^{3}{a}^{2}{d}^{3}+10\,\sqrt{-{\frac{b}{a}}}{x}^{3}abc{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}{b}^{2}{c}^{2}d+6\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}-8\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d+2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}+3\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){a}^{2}c{d}^{2}+7\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) ab{c}^{2}d-2\,\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ){b}^{2}{c}^{3}+6\,\sqrt{-{\frac{b}{a}}}x{a}^{2}c{d}^{2}+\sqrt{-{\frac{b}{a}}}xab{c}^{2}d \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(3/2)*(d*x^2+c)^(1/2),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+9*(-b/a)^(1/2)*
x^5*a*b*d^3+4*(-b/a)^(1/2)*x^5*b^2*c*d^2+6*(-b/a)^(1/2)*x^3*a^2*d^3+10*(-b/a)^(1
/2)*x^3*a*b*c*d^2+(-b/a)^(1/2)*x^3*b^2*c^2*d+6*((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^2-8*((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+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))*b^2
*c^3+3*((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-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellip
ticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b^2*c^3+6*(-b/a)^(1/2)*x*a^2*c*d^2+(-b/a)^(
1/2)*x*a*b*c^2*d)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/d^2/(-b/a)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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