3.831 \(\int \frac{\left (a+b x^2\right )^2 \sqrt{c+d x^2}}{x^{15/2}} \, dx\)

Optimal. Leaf size=441 \[ -\frac{2 \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^2 x^{5/2}}+\frac{2 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \sqrt{x} \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]

[Out]

(-2*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*Sqrt[c + d*x^2])/(195*c^2*x^(5/2)) - (
4*d*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*Sqrt[c + d*x^2])/(195*c^3*Sqrt[x]) + (
4*d^(3/2)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*Sqrt[x]*Sqrt[c + d*x^2])/(195*c^
3*(Sqrt[c] + Sqrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(13*c*x^(13/2)) - (2*a*(26*
b*c - 7*a*d)*(c + d*x^2)^(3/2))/(117*c^2*x^(9/2)) - (4*d^(5/4)*(39*b^2*c^2 - 26*
a*b*c*d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x
)^2]*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(195*c^(11/4)*Sqrt[c +
d*x^2]) + (2*d^(5/4)*(39*b^2*c^2 - 26*a*b*c*d + 7*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])/(195*c^(11/4)*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.887275, antiderivative size = 437, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{2 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d^{5/4} \left (\sqrt{c}+\sqrt{d} x\right ) \sqrt{\frac{c+d x^2}{\left (\sqrt{c}+\sqrt{d} x\right )^2}} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}}\right )|\frac{1}{2}\right )}{195 c^{11/4} \sqrt{c+d x^2}}-\frac{4 d \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \sqrt{x}}+\frac{4 d^{3/2} \sqrt{x} \sqrt{c+d x^2} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right )}{195 c^3 \left (\sqrt{c}+\sqrt{d} x\right )}-\frac{2 a^2 \left (c+d x^2\right )^{3/2}}{13 c x^{13/2}}-\frac{2 \sqrt{c+d x^2} \left (39 b^2-\frac{a d (26 b c-7 a d)}{c^2}\right )}{195 x^{5/2}}-\frac{2 a \left (c+d x^2\right )^{3/2} (26 b c-7 a d)}{117 c^2 x^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(15/2),x]

[Out]

(-2*(39*b^2 - (a*d*(26*b*c - 7*a*d))/c^2)*Sqrt[c + d*x^2])/(195*x^(5/2)) - (4*d*
(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*Sqrt[c + d*x^2])/(195*c^3*Sqrt[x]) + (4*d^
(3/2)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*Sqrt[x]*Sqrt[c + d*x^2])/(195*c^3*(S
qrt[c] + Sqrt[d]*x)) - (2*a^2*(c + d*x^2)^(3/2))/(13*c*x^(13/2)) - (2*a*(26*b*c
- 7*a*d)*(c + d*x^2)^(3/2))/(117*c^2*x^(9/2)) - (4*d^(5/4)*(39*b^2*c^2 - 26*a*b*
c*d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sqrt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]
*EllipticE[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(1/4)], 1/2])/(195*c^(11/4)*Sqrt[c + d*x
^2]) + (2*d^(5/4)*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*(Sqrt[c] + Sqrt[d]*x)*Sq
rt[(c + d*x^2)/(Sqrt[c] + Sqrt[d]*x)^2]*EllipticF[2*ArcTan[(d^(1/4)*Sqrt[x])/c^(
1/4)], 1/2])/(195*c^(11/4)*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 74.4561, size = 415, normalized size = 0.94 \[ - \frac{2 a^{2} \left (c + d x^{2}\right )^{\frac{3}{2}}}{13 c x^{\frac{13}{2}}} + \frac{2 a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (7 a d - 26 b c\right )}{117 c^{2} x^{\frac{9}{2}}} - \frac{2 \sqrt{c + d x^{2}} \left (a d \left (7 a d - 26 b c\right ) + 39 b^{2} c^{2}\right )}{195 c^{2} x^{\frac{5}{2}}} + \frac{4 d^{\frac{3}{2}} \sqrt{x} \sqrt{c + d x^{2}} \left (a d \left (7 a d - 26 b c\right ) + 39 b^{2} c^{2}\right )}{195 c^{3} \left (\sqrt{c} + \sqrt{d} x\right )} - \frac{4 d \sqrt{c + d x^{2}} \left (a d \left (7 a d - 26 b c\right ) + 39 b^{2} c^{2}\right )}{195 c^{3} \sqrt{x}} - \frac{4 d^{\frac{5}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (7 a d - 26 b c\right ) + 39 b^{2} c^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{195 c^{\frac{11}{4}} \sqrt{c + d x^{2}}} + \frac{2 d^{\frac{5}{4}} \sqrt{\frac{c + d x^{2}}{\left (\sqrt{c} + \sqrt{d} x\right )^{2}}} \left (\sqrt{c} + \sqrt{d} x\right ) \left (a d \left (7 a d - 26 b c\right ) + 39 b^{2} c^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{d} \sqrt{x}}{\sqrt [4]{c}} \right )}\middle | \frac{1}{2}\right )}{195 c^{\frac{11}{4}} \sqrt{c + d x^{2}}} \]

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**(15/2),x)

[Out]

-2*a**2*(c + d*x**2)**(3/2)/(13*c*x**(13/2)) + 2*a*(c + d*x**2)**(3/2)*(7*a*d -
26*b*c)/(117*c**2*x**(9/2)) - 2*sqrt(c + d*x**2)*(a*d*(7*a*d - 26*b*c) + 39*b**2
*c**2)/(195*c**2*x**(5/2)) + 4*d**(3/2)*sqrt(x)*sqrt(c + d*x**2)*(a*d*(7*a*d - 2
6*b*c) + 39*b**2*c**2)/(195*c**3*(sqrt(c) + sqrt(d)*x)) - 4*d*sqrt(c + d*x**2)*(
a*d*(7*a*d - 26*b*c) + 39*b**2*c**2)/(195*c**3*sqrt(x)) - 4*d**(5/4)*sqrt((c + d
*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(c) + sqrt(d)*x)*(a*d*(7*a*d - 26*b*c) + 3
9*b**2*c**2)*elliptic_e(2*atan(d**(1/4)*sqrt(x)/c**(1/4)), 1/2)/(195*c**(11/4)*s
qrt(c + d*x**2)) + 2*d**(5/4)*sqrt((c + d*x**2)/(sqrt(c) + sqrt(d)*x)**2)*(sqrt(
c) + sqrt(d)*x)*(a*d*(7*a*d - 26*b*c) + 39*b**2*c**2)*elliptic_f(2*atan(d**(1/4)
*sqrt(x)/c**(1/4)), 1/2)/(195*c**(11/4)*sqrt(c + d*x**2))

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Mathematica [C]  time = 0.701818, size = 241, normalized size = 0.55 \[ \frac{2 \left (\left (-c-d x^2\right ) \left (a^2 \left (45 c^3+10 c^2 d x^2-14 c d^2 x^4+42 d^3 x^6\right )+26 a b c x^2 \left (5 c^2+2 c d x^2-6 d^2 x^4\right )+117 b^2 c^2 x^4 \left (c+2 d x^2\right )\right )+\frac{6 i d^2 x^8 \sqrt{\frac{d x^2}{c}+1} \left (7 a^2 d^2-26 a b c d+39 b^2 c^2\right ) \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{d} x}{\sqrt{c}}}\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{d} x}{\sqrt{c}}\right )^{3/2}}\right )}{585 c^3 x^{13/2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^2)^2*Sqrt[c + d*x^2])/x^(15/2),x]

[Out]

(2*((-c - d*x^2)*(117*b^2*c^2*x^4*(c + 2*d*x^2) + 26*a*b*c*x^2*(5*c^2 + 2*c*d*x^
2 - 6*d^2*x^4) + a^2*(45*c^3 + 10*c^2*d*x^2 - 14*c*d^2*x^4 + 42*d^3*x^6)) + ((6*
I)*d^2*(39*b^2*c^2 - 26*a*b*c*d + 7*a^2*d^2)*x^8*Sqrt[1 + (d*x^2)/c]*(EllipticE[
I*ArcSinh[Sqrt[(I*Sqrt[d]*x)/Sqrt[c]]], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[d
]*x)/Sqrt[c]]], -1]))/((I*Sqrt[d]*x)/Sqrt[c])^(3/2)))/(585*c^3*x^(13/2)*Sqrt[c +
d*x^2])

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Maple [A]  time = 0.054, size = 706, normalized size = 1.6 \[{\frac{2}{585\,{c}^{3}} \left ( 42\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{a}^{2}c{d}^{3}-156\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}ab{c}^{2}{d}^{2}+234\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticE} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{b}^{2}{c}^{3}d-21\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{a}^{2}c{d}^{3}+78\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}ab{c}^{2}{d}^{2}-117\,\sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{2}\sqrt{{\frac{-dx+\sqrt{-cd}}{\sqrt{-cd}}}}\sqrt{-{\frac{dx}{\sqrt{-cd}}}}{\it EllipticF} \left ( \sqrt{{\frac{dx+\sqrt{-cd}}{\sqrt{-cd}}}},1/2\,\sqrt{2} \right ){x}^{6}{b}^{2}{c}^{3}d-42\,{x}^{8}{a}^{2}{d}^{4}+156\,{x}^{8}abc{d}^{3}-234\,{x}^{8}{b}^{2}{c}^{2}{d}^{2}-28\,{x}^{6}{a}^{2}c{d}^{3}+104\,{x}^{6}ab{c}^{2}{d}^{2}-351\,{x}^{6}{b}^{2}{c}^{3}d+4\,{x}^{4}{a}^{2}{c}^{2}{d}^{2}-182\,{x}^{4}ab{c}^{3}d-117\,{x}^{4}{b}^{2}{c}^{4}-55\,{x}^{2}{a}^{2}{c}^{3}d-130\,{x}^{2}ab{c}^{4}-45\,{a}^{2}{c}^{4} \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}{x}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2*(d*x^2+c)^(1/2)/x^(15/2),x)

[Out]

2/585*(42*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d
)^(1/2))^(1/2),1/2*2^(1/2))*x^6*a^2*c*d^3-156*((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)
*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^6*a*b*c^2*d^2+
234*((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)*EllipticE(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2
))^(1/2),1/2*2^(1/2))*x^6*b^2*c^3*d-21*((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)*Ellipt
icF(((d*x+(-c*d)^(1/2))/(-c*d)^(1/2))^(1/2),1/2*2^(1/2))*x^6*a^2*c*d^3+78*((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))*x^6*a*b*c^2*d^2-117*((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))*x^6*b^2*c^3*d-42*x^8*a^2*d^4+1
56*x^8*a*b*c*d^3-234*x^8*b^2*c^2*d^2-28*x^6*a^2*c*d^3+104*x^6*a*b*c^2*d^2-351*x^
6*b^2*c^3*d+4*x^4*a^2*c^2*d^2-182*x^4*a*b*c^3*d-117*x^4*b^2*c^4-55*x^2*a^2*c^3*d
-130*x^2*a*b*c^4-45*a^2*c^4)/(d*x^2+c)^(1/2)/x^(13/2)/c^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(15/2),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(15/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(15/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4 + 2*a*b*x^2 + a^2)*sqrt(d*x^2 + c)/x^(15/2), 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**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{2} + a\right )}^{2} \sqrt{d x^{2} + c}}{x^{\frac{15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(15/2),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^2*sqrt(d*x^2 + c)/x^(15/2), x)