3.1118 \(\int \frac {c+d x^2}{(e x)^{3/2} (a+b x^2)^{7/4}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}} \]

[Out]

-2/3*(-a*d+4*b*c)*(e*x)^(3/2)/a^2/e^3/(b*x^2+a)^(3/4)-2*c/a/e/(b*x^2+a)^(3/4)/(e*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {453, 264} \[ -\frac {2 (e x)^{3/2} (4 b c-a d)}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]

[Out]

(-2*c)/(a*e*Sqrt[e*x]*(a + b*x^2)^(3/4)) - (2*(4*b*c - a*d)*(e*x)^(3/2))/(3*a^2*e^3*(a + b*x^2)^(3/4))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rubi steps

\begin {align*} \int \frac {c+d x^2}{(e x)^{3/2} \left (a+b x^2\right )^{7/4}} \, dx &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {(4 b c-a d) \int \frac {\sqrt {e x}}{\left (a+b x^2\right )^{7/4}} \, dx}{a e^2}\\ &=-\frac {2 c}{a e \sqrt {e x} \left (a+b x^2\right )^{3/4}}-\frac {2 (4 b c-a d) (e x)^{3/2}}{3 a^2 e^3 \left (a+b x^2\right )^{3/4}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 0.68 \[ \frac {2 x \left (-3 a c+a d x^2-4 b c x^2\right )}{3 a^2 (e x)^{3/2} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x]

[Out]

(2*x*(-3*a*c - 4*b*c*x^2 + a*d*x^2))/(3*a^2*(e*x)^(3/2)*(a + b*x^2)^(3/4))

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fricas [A]  time = 1.37, size = 56, normalized size = 0.86 \[ -\frac {2 \, {\left ({\left (4 \, b c - a d\right )} x^{2} + 3 \, a c\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{3 \, {\left (a^{2} b e^{2} x^{3} + a^{3} e^{2} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(7/4),x, algorithm="fricas")

[Out]

-2/3*((4*b*c - a*d)*x^2 + 3*a*c)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(a^2*b*e^2*x^3 + a^3*e^2*x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(7/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(3/2)), x)

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maple [A]  time = 0.01, size = 40, normalized size = 0.62 \[ -\frac {2 \left (-a d \,x^{2}+4 b c \,x^{2}+3 a c \right ) x}{3 \left (b \,x^{2}+a \right )^{\frac {3}{4}} \left (e x \right )^{\frac {3}{2}} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(7/4),x)

[Out]

-2/3*x*(-a*d*x^2+4*b*c*x^2+3*a*c)/(b*x^2+a)^(3/4)/a^2/(e*x)^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {7}{4}} \left (e x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(3/2)/(b*x^2+a)^(7/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(7/4)*(e*x)^(3/2)), x)

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mupad [B]  time = 1.18, size = 69, normalized size = 1.06 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{a\,b\,e}-\frac {x^2\,\left (2\,a\,d-8\,b\,c\right )}{3\,a^2\,b\,e}\right )}{x^2\,\sqrt {e\,x}+\frac {a\,\sqrt {e\,x}}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/((e*x)^(3/2)*(a + b*x^2)^(7/4)),x)

[Out]

-((a + b*x^2)^(1/4)*((2*c)/(a*b*e) - (x^2*(2*a*d - 8*b*c))/(3*a^2*b*e)))/(x^2*(e*x)^(1/2) + (a*(e*x)^(1/2))/b)

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sympy [A]  time = 79.59, size = 119, normalized size = 1.83 \[ c \left (\frac {3 \Gamma \left (- \frac {1}{4}\right )}{8 a b^{\frac {3}{4}} e^{\frac {3}{2}} x^{2} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt [4]{b} \Gamma \left (- \frac {1}{4}\right )}{2 a^{2} e^{\frac {3}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )}\right ) + \frac {d \Gamma \left (\frac {3}{4}\right )}{2 a b^{\frac {3}{4}} e^{\frac {3}{2}} \left (\frac {a}{b x^{2}} + 1\right )^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/(e*x)**(3/2)/(b*x**2+a)**(7/4),x)

[Out]

c*(3*gamma(-1/4)/(8*a*b**(3/4)*e**(3/2)*x**2*(a/(b*x**2) + 1)**(3/4)*gamma(7/4)) + b**(1/4)*gamma(-1/4)/(2*a**
2*e**(3/2)*(a/(b*x**2) + 1)**(3/4)*gamma(7/4))) + d*gamma(3/4)/(2*a*b**(3/4)*e**(3/2)*(a/(b*x**2) + 1)**(3/4)*
gamma(7/4))

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