3.1126 \(\int \frac{(e x)^{7/2} \left (c+d x^2\right )}{\left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=221 \[ \frac{e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac{e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac{e^3 \sqrt{e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

[Out]

(2*(b*c - a*d)*(e*x)^(9/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((4*b*c - 9*a*d)*e^3*S
qrt[e*x])/(2*b^3*(a + b*x^2)^(1/4)) - ((4*b*c - 9*a*d)*e*(e*x)^(5/2))/(10*a*b^2*
(a + b*x^2)^(1/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt[e
]*(a + b*x^2)^(1/4))])/(4*b^(13/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTanh[(b^(1/4)*
Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(13/4))

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Rubi [A]  time = 0.391884, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{e^{7/2} (4 b c-9 a d) \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}+\frac{e^{7/2} (4 b c-9 a d) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a+b x^2}}\right )}{4 b^{13/4}}-\frac{e^3 \sqrt{e x} (4 b c-9 a d)}{2 b^3 \sqrt [4]{a+b x^2}}-\frac{e (e x)^{5/2} (4 b c-9 a d)}{10 a b^2 \sqrt [4]{a+b x^2}}+\frac{2 (e x)^{9/2} (b c-a d)}{5 a b e \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(2*(b*c - a*d)*(e*x)^(9/2))/(5*a*b*e*(a + b*x^2)^(5/4)) - ((4*b*c - 9*a*d)*e^3*S
qrt[e*x])/(2*b^3*(a + b*x^2)^(1/4)) - ((4*b*c - 9*a*d)*e*(e*x)^(5/2))/(10*a*b^2*
(a + b*x^2)^(1/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTan[(b^(1/4)*Sqrt[e*x])/(Sqrt[e
]*(a + b*x^2)^(1/4))])/(4*b^(13/4)) + ((4*b*c - 9*a*d)*e^(7/2)*ArcTanh[(b^(1/4)*
Sqrt[e*x])/(Sqrt[e]*(a + b*x^2)^(1/4))])/(4*b^(13/4))

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Rubi in Sympy [A]  time = 41.5549, size = 194, normalized size = 0.88 \[ \frac{d \left (e x\right )^{\frac{9}{2}}}{2 b e \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e \left (e x\right )^{\frac{5}{2}} \left (9 a d - 4 b c\right )}{10 b^{2} \left (a + b x^{2}\right )^{\frac{5}{4}}} + \frac{e^{3} \sqrt{e x} \left (9 a d - 4 b c\right )}{2 b^{3} \sqrt [4]{a + b x^{2}}} - \frac{e^{\frac{7}{2}} \left (9 a d - 4 b c\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{13}{4}}} - \frac{e^{\frac{7}{2}} \left (9 a d - 4 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{e x}}{\sqrt{e} \sqrt [4]{a + b x^{2}}} \right )}}{4 b^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

d*(e*x)**(9/2)/(2*b*e*(a + b*x**2)**(5/4)) + e*(e*x)**(5/2)*(9*a*d - 4*b*c)/(10*
b**2*(a + b*x**2)**(5/4)) + e**3*sqrt(e*x)*(9*a*d - 4*b*c)/(2*b**3*(a + b*x**2)*
*(1/4)) - e**(7/2)*(9*a*d - 4*b*c)*atan(b**(1/4)*sqrt(e*x)/(sqrt(e)*(a + b*x**2)
**(1/4)))/(4*b**(13/4)) - e**(7/2)*(9*a*d - 4*b*c)*atanh(b**(1/4)*sqrt(e*x)/(sqr
t(e)*(a + b*x**2)**(1/4)))/(4*b**(13/4))

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Mathematica [C]  time = 0.186947, size = 116, normalized size = 0.52 \[ \frac{e^3 \sqrt{e x} \left (45 a^2 d+5 \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} (4 b c-9 a d) \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{b x^2}{a}\right )+a b \left (54 d x^2-20 c\right )+b^2 x^2 \left (5 d x^2-24 c\right )\right )}{10 b^3 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[((e*x)^(7/2)*(c + d*x^2))/(a + b*x^2)^(9/4),x]

[Out]

(e^3*Sqrt[e*x]*(45*a^2*d + b^2*x^2*(-24*c + 5*d*x^2) + a*b*(-20*c + 54*d*x^2) +
5*(4*b*c - 9*a*d)*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 1/4,
5/4, -((b*x^2)/a)]))/(10*b^3*(a + b*x^2)^(5/4))

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int{(d{x}^{2}+c) \left ( ex \right ) ^{{\frac{7}{2}}} \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

[Out]

int((e*x)^(7/2)*(d*x^2+c)/(b*x^2+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4), x)

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Fricas [A]  time = 0.284408, size = 1108, normalized size = 5.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4),x, algorithm="fricas")

[Out]

1/40*(4*(5*b^2*d*e^3*x^4 - 6*(4*b^2*c - 9*a*b*d)*e^3*x^2 - 5*(4*a*b*c - 9*a^2*d)
*e^3)*(b*x^2 + a)^(3/4)*sqrt(e*x) + 20*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*((256*b
^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*
d^4)*e^14/b^13)^(1/4)*arctan(-(b^4*x^2 + a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d
+ 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)/((b
*x^2 + a)^(3/4)*(4*b*c - 9*a*d)*sqrt(e*x)*e^3 - (b*x^2 + a)*sqrt(((16*b^2*c^2 -
72*a*b*c*d + 81*a^2*d^2)*sqrt(b*x^2 + a)*e^7*x + (b^7*x^2 + a*b^6)*sqrt((256*b^4
*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^
4)*e^14/b^13))/(b*x^2 + a)))) + 5*(b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*((256*b^4*c^
4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*
e^14/b^13)^(1/4)*log(-((b*x^2 + a)^(3/4)*(4*b*c - 9*a*d)*sqrt(e*x)*e^3 + (b^4*x^
2 + a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*b
*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4))/(b*x^2 + a)) - 5*(b^5*x^4 + 2*a*b^4*x^2
+ a^2*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b^2*c^2*d^2 - 11664*a^3*
b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4)*log(-((b*x^2 + a)^(3/4)*(4*b*c - 9*a*d)
*sqrt(e*x)*e^3 - (b^4*x^2 + a*b^3)*((256*b^4*c^4 - 2304*a*b^3*c^3*d + 7776*a^2*b
^2*c^2*d^2 - 11664*a^3*b*c*d^3 + 6561*a^4*d^4)*e^14/b^13)^(1/4))/(b*x^2 + a)))/(
b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)

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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((e*x)**(7/2)*(d*x**2+c)/(b*x**2+a)**(9/4),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (d x^{2} + c\right )} \left (e x\right )^{\frac{7}{2}}}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)*(e*x)^(7/2)/(b*x^2 + a)^(9/4), x)