3.1098 \(\int \frac {c+d x^2}{(e x)^{15/2} (a+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=141 \[ \frac {64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac {16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac {2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]

[Out]

-2/13*c*(b*x^2+a)^(1/4)/a/e/(e*x)^(13/2)+2/13*(-13*a*d+12*b*c)*(b*x^2+a)^(1/4)/a^2/e^3/(e*x)^(9/2)-16/65*(-13*
a*d+12*b*c)*(b*x^2+a)^(5/4)/a^3/e^3/(e*x)^(9/2)+64/585*(-13*a*d+12*b*c)*(b*x^2+a)^(9/4)/a^4/e^3/(e*x)^(9/2)

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Rubi [A]  time = 0.07, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {453, 273, 264} \[ \frac {64 \left (a+b x^2\right )^{9/4} (12 b c-13 a d)}{585 a^4 e^3 (e x)^{9/2}}-\frac {16 \left (a+b x^2\right )^{5/4} (12 b c-13 a d)}{65 a^3 e^3 (e x)^{9/2}}+\frac {2 \sqrt [4]{a+b x^2} (12 b c-13 a d)}{13 a^2 e^3 (e x)^{9/2}}-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*c*(a + b*x^2)^(1/4))/(13*a*e*(e*x)^(13/2)) + (2*(12*b*c - 13*a*d)*(a + b*x^2)^(1/4))/(13*a^2*e^3*(e*x)^(9/
2)) - (16*(12*b*c - 13*a*d)*(a + b*x^2)^(5/4))/(65*a^3*e^3*(e*x)^(9/2)) + (64*(12*b*c - 13*a*d)*(a + b*x^2)^(9
/4))/(585*a^4*e^3*(e*x)^(9/2))

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 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -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)^{15/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}-\frac {(12 b c-13 a d) \int \frac {1}{(e x)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx}{13 a e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}+\frac {(8 (12 b c-13 a d)) \int \frac {\sqrt [4]{a+b x^2}}{(e x)^{11/2}} \, dx}{13 a^2 e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac {16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}-\frac {(32 (12 b c-13 a d)) \int \frac {\left (a+b x^2\right )^{5/4}}{(e x)^{11/2}} \, dx}{65 a^3 e^2}\\ &=-\frac {2 c \sqrt [4]{a+b x^2}}{13 a e (e x)^{13/2}}+\frac {2 (12 b c-13 a d) \sqrt [4]{a+b x^2}}{13 a^2 e^3 (e x)^{9/2}}-\frac {16 (12 b c-13 a d) \left (a+b x^2\right )^{5/4}}{65 a^3 e^3 (e x)^{9/2}}+\frac {64 (12 b c-13 a d) \left (a+b x^2\right )^{9/4}}{585 a^4 e^3 (e x)^{9/2}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 94, normalized size = 0.67 \[ -\frac {2 \sqrt {e x} \sqrt [4]{a+b x^2} \left (5 a^3 \left (9 c+13 d x^2\right )-4 a^2 b x^2 \left (15 c+26 d x^2\right )+32 a b^2 x^4 \left (3 c+13 d x^2\right )-384 b^3 c x^6\right )}{585 a^4 e^8 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*Sqrt[e*x]*(a + b*x^2)^(1/4)*(-384*b^3*c*x^6 + 32*a*b^2*x^4*(3*c + 13*d*x^2) + 5*a^3*(9*c + 13*d*x^2) - 4*a
^2*b*x^2*(15*c + 26*d*x^2)))/(585*a^4*e^8*x^7)

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fricas [A]  time = 1.24, size = 90, normalized size = 0.64 \[ \frac {2 \, {\left (32 \, {\left (12 \, b^{3} c - 13 \, a b^{2} d\right )} x^{6} - 8 \, {\left (12 \, a b^{2} c - 13 \, a^{2} b d\right )} x^{4} - 45 \, a^{3} c + 5 \, {\left (12 \, a^{2} b c - 13 \, a^{3} d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {e x}}{585 \, a^{4} e^{8} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

2/585*(32*(12*b^3*c - 13*a*b^2*d)*x^6 - 8*(12*a*b^2*c - 13*a^2*b*d)*x^4 - 45*a^3*c + 5*(12*a^2*b*c - 13*a^3*d)
*x^2)*(b*x^2 + a)^(1/4)*sqrt(e*x)/(a^4*e^8*x^7)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)), x)

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maple [A]  time = 0.01, size = 86, normalized size = 0.61 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (416 a \,b^{2} d \,x^{6}-384 b^{3} c \,x^{6}-104 a^{2} b d \,x^{4}+96 a \,b^{2} c \,x^{4}+65 a^{3} d \,x^{2}-60 a^{2} b c \,x^{2}+45 c \,a^{3}\right ) x}{585 \left (e x \right )^{\frac {15}{2}} a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x)

[Out]

-2/585*(b*x^2+a)^(1/4)*x*(416*a*b^2*d*x^6-384*b^3*c*x^6-104*a^2*b*d*x^4+96*a*b^2*c*x^4+65*a^3*d*x^2-60*a^2*b*c
*x^2+45*a^3*c)/a^4/(e*x)^(15/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d x^{2} + c}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (e x\right )^{\frac {15}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^2+c)/(e*x)^(15/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate((d*x^2 + c)/((b*x^2 + a)^(3/4)*(e*x)^(15/2)), x)

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mupad [B]  time = 1.26, size = 100, normalized size = 0.71 \[ -\frac {{\left (b\,x^2+a\right )}^{1/4}\,\left (\frac {2\,c}{13\,a\,e^7}+\frac {x^2\,\left (130\,a^3\,d-120\,a^2\,b\,c\right )}{585\,a^4\,e^7}-\frac {x^6\,\left (768\,b^3\,c-832\,a\,b^2\,d\right )}{585\,a^4\,e^7}-\frac {16\,b\,x^4\,\left (13\,a\,d-12\,b\,c\right )}{585\,a^3\,e^7}\right )}{x^6\,\sqrt {e\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^2)/((e*x)^(15/2)*(a + b*x^2)^(3/4)),x)

[Out]

-((a + b*x^2)^(1/4)*((2*c)/(13*a*e^7) + (x^2*(130*a^3*d - 120*a^2*b*c))/(585*a^4*e^7) - (x^6*(768*b^3*c - 832*
a*b^2*d))/(585*a^4*e^7) - (16*b*x^4*(13*a*d - 12*b*c))/(585*a^3*e^7)))/(x^6*(e*x)^(1/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x**2+c)/(e*x)**(15/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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