3.3.10 \(\int (d+e x)^m \log (c (a+\frac {b}{x^2})^p) \, dx\) [210]

Optimal. Leaf size=257 \[ \frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e \left (\sqrt {-a} d-\sqrt {b} e\right ) (1+m) (2+m)}+\frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e \left (\sqrt {-a} d+\sqrt {b} e\right ) (1+m) (2+m)}-\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)} \]

[Out]

-2*p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],1+e*x/d)/d/e/(m^2+3*m+2)+(e*x+d)^(1+m)*ln(c*(a+b/x^2)^p)/e/(1+m)+p
*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],(e*x+d)*(-a)^(1/2)/(d*(-a)^(1/2)-e*b^(1/2)))*(-a)^(1/2)/e/(1+m)/(2+m)/
(d*(-a)^(1/2)-e*b^(1/2))+p*(e*x+d)^(2+m)*hypergeom([1, 2+m],[3+m],(e*x+d)*(-a)^(1/2)/(d*(-a)^(1/2)+e*b^(1/2)))
*(-a)^(1/2)/e/(1+m)/(2+m)/(d*(-a)^(1/2)+e*b^(1/2))

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Rubi [A]
time = 0.36, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2513, 1584, 975, 67, 845, 70} \begin {gather*} \frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (m+1)}+\frac {\sqrt {-a} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} d-\sqrt {b} e\right )}+\frac {\sqrt {-a} p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e (m+1) (m+2) \left (\sqrt {-a} d+\sqrt {b} e\right )}-\frac {2 p (d+e x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {e x}{d}+1\right )}{d e \left (m^2+3 m+2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*Log[c*(a + b/x^2)^p],x]

[Out]

(Sqrt[-a]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)
])/(e*(Sqrt[-a]*d - Sqrt[b]*e)*(1 + m)*(2 + m)) + (Sqrt[-a]*p*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3
+ m, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)])/(e*(Sqrt[-a]*d + Sqrt[b]*e)*(1 + m)*(2 + m)) - (2*p*(d +
e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, 1 + (e*x)/d])/(d*e*(2 + 3*m + m^2)) + ((d + e*x)^(1 + m)*Log[c
*(a + b/x^2)^p])/(e*(1 + m))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 845

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m, (f + g*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !Ration
alQ[m]

Rule 975

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0])) &&  !(IGtQ[m, 0] || IGtQ[n, 0])

Rule 1584

Int[(x_)^(m_.)*((a_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[x^(m - 2*n
*p)*(d + e*x^n)^q*(c + a*x^(2*n))^p, x] /; FreeQ[{a, c, d, e, m, n, q}, x] && EqQ[mn2, -2*n] && IntegerQ[p]

Rule 2513

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Simp[(f
+ g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n)^p])/(g*(r + 1))), x] - Dist[b*e*n*(p/(g*(r + 1))), Int[x^(n - 1)*((f
+ g*x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x] && (IGtQ[r, 0] || RationalQ[n
]) && NeQ[r, -1]

Rubi steps

\begin {align*} \int (d+e x)^m \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \frac {(d+e x)^{1+m}}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \frac {(d+e x)^{1+m}}{x \left (b+a x^2\right )} \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 b p) \int \left (\frac {(d+e x)^{1+m}}{b x}-\frac {a x (d+e x)^{1+m}}{b \left (b+a x^2\right )}\right ) \, dx}{e (1+m)}\\ &=\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {(2 p) \int \frac {(d+e x)^{1+m}}{x} \, dx}{e (1+m)}-\frac {(2 a p) \int \frac {x (d+e x)^{1+m}}{b+a x^2} \, dx}{e (1+m)}\\ &=-\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}-\frac {(2 a p) \int \left (-\frac {\sqrt {-a} (d+e x)^{1+m}}{2 a \left (\sqrt {b}-\sqrt {-a} x\right )}+\frac {\sqrt {-a} (d+e x)^{1+m}}{2 a \left (\sqrt {b}+\sqrt {-a} x\right )}\right ) \, dx}{e (1+m)}\\ &=-\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}+\frac {\left (\sqrt {-a} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {b}-\sqrt {-a} x} \, dx}{e (1+m)}-\frac {\left (\sqrt {-a} p\right ) \int \frac {(d+e x)^{1+m}}{\sqrt {b}+\sqrt {-a} x} \, dx}{e (1+m)}\\ &=\frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e \left (\sqrt {-a} d-\sqrt {b} e\right ) (1+m) (2+m)}+\frac {\sqrt {-a} p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e \left (\sqrt {-a} d+\sqrt {b} e\right ) (1+m) (2+m)}-\frac {2 p (d+e x)^{2+m} \, _2F_1\left (1,2+m;3+m;1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e (1+m)}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.70, size = 310, normalized size = 1.21 \begin {gather*} \frac {(d+e x)^m \left (2 d p \left (1+\frac {d}{e x}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {d}{e x}\right )-\frac {\left (\sqrt {a} d+i \sqrt {b} e\right ) p \left (\frac {\sqrt {a} (d+e x)}{e \left (-i \sqrt {b}+\sqrt {a} x\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac {\sqrt {a} d+i \sqrt {b} e}{i \sqrt {b} e-\sqrt {a} e x}\right )}{\sqrt {a}}-\frac {\left (\sqrt {a} d-i \sqrt {b} e\right ) p \left (\frac {\sqrt {a} (d+e x)}{e \left (i \sqrt {b}+\sqrt {a} x\right )}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac {\sqrt {a} d-i \sqrt {b} e}{i \sqrt {b} e+\sqrt {a} e x}\right )}{\sqrt {a}}+m (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )\right )}{e m (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^m*Log[c*(a + b/x^2)^p],x]

[Out]

((d + e*x)^m*((2*d*p*Hypergeometric2F1[-m, -m, 1 - m, -(d/(e*x))])/(1 + d/(e*x))^m - ((Sqrt[a]*d + I*Sqrt[b]*e
)*p*Hypergeometric2F1[-m, -m, 1 - m, (Sqrt[a]*d + I*Sqrt[b]*e)/(I*Sqrt[b]*e - Sqrt[a]*e*x)])/(Sqrt[a]*((Sqrt[a
]*(d + e*x))/(e*((-I)*Sqrt[b] + Sqrt[a]*x)))^m) - ((Sqrt[a]*d - I*Sqrt[b]*e)*p*Hypergeometric2F1[-m, -m, 1 - m
, -((Sqrt[a]*d - I*Sqrt[b]*e)/(I*Sqrt[b]*e + Sqrt[a]*e*x))])/(Sqrt[a]*((Sqrt[a]*(d + e*x))/(e*(I*Sqrt[b] + Sqr
t[a]*x)))^m) + m*(d + e*x)*Log[c*(a + b/x^2)^p]))/(e*m*(1 + m))

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Maple [F]
time = 0.25, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m*ln(c*(a+b/x^2)^p),x)

[Out]

int((e*x+d)^m*ln(c*(a+b/x^2)^p),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="maxima")

[Out]

(p*x*e + d*p)*e^(m*log(x*e + d) - 1)*log(a*x^2 + b)/(m + 1) - integrate(-(((m + 1)*log(c) - 2*p)*a*x^2*e - 2*a
*d*p*x + b*(m + 1)*e*log(c) - 2*(a*(m + 1)*x^2*e + b*(m + 1)*e)*log(x^p))*(x*e + d)^m/(a*(m + 1)*x^2*e + b*(m
+ 1)*e), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="fricas")

[Out]

integral((x*e + d)^m*log(c*((a*x^2 + b)/x^2)^p), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**m*ln(c*(a+b/x**2)**p),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^m*log(c*(a+b/x^2)^p),x, algorithm="giac")

[Out]

integrate((x*e + d)^m*log((a + b/x^2)^p*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )\,{\left (d+e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x^2)^p)*(d + e*x)^m,x)

[Out]

int(log(c*(a + b/x^2)^p)*(d + e*x)^m, x)

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