3.830 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{13/2}} \, dx\)

Optimal. Leaf size=217 \[ -\frac{2 \sqrt{c+d x^2} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right )}{231 c^2 x^{3/2}}+\frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]

[Out]

(-2*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*Sqrt[c + d*x^2])/(231*c^2*x^(3/2)) - (
2*a^2*(c + d*x^2)^(3/2))/(11*c*x^(11/2)) - (2*a*(22*b*c - 5*a*d)*(c + d*x^2)^(3/
2))/(77*c^2*x^(7/2)) + (2*d^(3/4)*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*(Sqrt[c]
+ Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1
/4)*Sqrt[x])/c^(1/4)], 1/2])/(231*c^(9/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.455355, antiderivative size = 213, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{2 d^{3/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{c+d x^2}}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{11 c x^{11/2}}-\frac{2 \sqrt{c+d x^2} \left (77 b^2-\frac{a d (22 b c-5 a d)}{c^2}\right )}{231 x^{3/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (22 b c-5 a d)}{77 c^2 x^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]

[Out]

(-2*(77*b^2 - (a*d*(22*b*c - 5*a*d))/c^2)*Sqrt[c + d*x^2])/(231*x^(3/2)) - (2*a^
2*(c + d*x^2)^(3/2))/(11*c*x^(11/2)) - (2*a*(22*b*c - 5*a*d)*(c + d*x^2)^(3/2))/
(77*c^2*x^(7/2)) + (2*d^(3/4)*(77*b^2*c^2 - 22*a*b*c*d + 5*a^2*d^2)*(Sqrt[c] + S
qrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*
Sqrt[x])/c^(1/4)], 1/2])/(231*c^(9/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 37.8611, size = 204, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{11 c x^{\frac{11}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (5 a d - 22 b c\right )}{77 c^{2} x^{\frac{7}{2}}} - \frac{2 \sqrt{c + d x^{2}} \left (a d \left (5 a d - 22 b c\right ) + 77 b^{2} c^{2}\right )}{231 c^{2} x^{\frac{3}{2}}} + \frac{2 d^{\frac{3}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (5 a d - 22 b c\right ) + 77 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{9}{4}} \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(13/2),x)

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(11*c*x**(11/2)) + 2*a*(c + d*x**2)**(3/2)*(5*a*d -
22*b*c)/(77*c**2*x**(7/2)) - 2*sqrt(c + d*x**2)*(a*d*(5*a*d - 22*b*c) + 77*b**2*
c**2)/(231*c**2*x**(3/2)) + 2*d**(3/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**
2)*(sqrt(c) + sqrt(d)*x)*(a*d*(5*a*d - 22*b*c) + 77*b**2*c**2)*elliptic_f(2*atan
(d**(1/4)*sqrt(x)/c**(1/4)), 1/2)/(231*c**(9/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.244865, size = 187, normalized size = 0.86 \[ -\frac{2 \sqrt{c+d x^2} \left (a^2 \left (21 c^2+6 c d x^2-10 d^2 x^4\right )+22 a b c x^2 \left (3 c+2 d x^2\right )+77 b^2 c^2 x^4\right )}{231 c^2 x^{11/2}}+\frac{4 i d x \sqrt{\frac{c}{d x^2}+1} \left (5 a^2 d^2-22 a b c d+77 b^2 c^2\right ) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{c}}{\sqrt{d}}}}{\sqrt{x}}\right )\right |-1\right )}{231 c^2 \sqrt{\frac{i \sqrt{c}}{\sqrt{d}}} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(13/2),x]

[Out]

(-2*Sqrt[c + d*x^2]*(77*b^2*c^2*x^4 + 22*a*b*c*x^2*(3*c + 2*d*x^2) + a^2*(21*c^2
+ 6*c*d*x^2 - 10*d^2*x^4)))/(231*c^2*x^(11/2)) + (((4*I)/231)*d*(77*b^2*c^2 - 2
2*a*b*c*d + 5*a^2*d^2)*Sqrt[1 + c/(d*x^2)]*x*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]
)/Sqrt[d]]/Sqrt[x]], -1])/(c^2*Sqrt[(I*Sqrt[c])/Sqrt[d]]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.049, size = 403, normalized size = 1.9 \[{\frac{2}{231\,{c}^{2}} \left ( 5\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{a}^{2}{d}^{2}-22\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}abcd+77\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ) \sqrt{-cd}{x}^{5}{b}^{2}{c}^{2}+10\,{x}^{6}{a}^{2}{d}^{3}-44\,{x}^{6}abc{d}^{2}-77\,{x}^{6}{b}^{2}{c}^{2}d+4\,{x}^{4}{a}^{2}c{d}^{2}-110\,{x}^{4}ab{c}^{2}d-77\,{x}^{4}{b}^{2}{c}^{3}-27\,{x}^{2}{a}^{2}{c}^{2}d-66\,{x}^{2}ab{c}^{3}-21\,{a}^{2}{c}^{3} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(13/2),x)

[Out]

2/231/(d*x^2+c)^(1/2)*(5*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+
(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*
d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*a^2*d^2-22*((d*x+(-c
*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*
(-x/(-c*d)^(1/2)*d)^(1/2)*EllipticF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*
2^(1/2))*(-c*d)^(1/2)*x^5*a*b*c*d+77*((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*2^(
1/2)*((-d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2)*(-x/(-c*d)^(1/2)*d)^(1/2)*Elliptic
F(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*(-c*d)^(1/2)*x^5*b^2*c^2+
10*x^6*a^2*d^3-44*x^6*a*b*c*d^2-77*x^6*b^2*c^2*d+4*x^4*a^2*c*d^2-110*x^4*a*b*c^2
*d-77*x^4*b^2*c^3-27*x^2*a^2*c^2*d-66*x^2*a*b*c^3-21*a^2*c^3)/x^(11/2)/c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(13/2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2*(d*x**2+c)**(1/2)/x**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(13/2), x)