Optimal. Leaf size=559 \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.731034, antiderivative size = 559, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {745, 837, 807, 727} \[ -\frac{c^2 \left (\sqrt{-a}-\sqrt{c} x\right ) \left (a+c x^2\right )^p (d+e x)^{-2 p-1} \left (3 a^2 e^4-6 a c d^2 e^2 (2 p+5)+c^2 d^4 \left (4 p^2+16 p+15\right )\right ) \left (-\frac{\left (\sqrt{-a}+\sqrt{c} x\right ) \left (\sqrt{-a} e+\sqrt{c} d\right )}{\left (\sqrt{-a}-\sqrt{c} x\right ) \left (\sqrt{c} d-\sqrt{-a} e\right )}\right )^{-p} \, _2F_1\left (-2 p-1,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{(2 p+1) (2 p+3) (2 p+5) \left (\sqrt{-a} e+\sqrt{c} d\right ) \left (a e^2+c d^2\right )^4}+\frac{c^2 d e (p+3) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+1)} \left (a e^2 (5 p+8)-c d^2 \left (2 p^2+7 p+8\right )\right )}{(p+1) (p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^4}+\frac{c e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-3} \left (3 a e^2 (p+2)-c d^2 \left (2 p^2+11 p+18\right )\right )}{(p+2) (2 p+3) (2 p+5) \left (a e^2+c d^2\right )^3}-\frac{e \left (a+c x^2\right )^{p+1} (d+e x)^{-2 p-5}}{(2 p+5) \left (a e^2+c d^2\right )}-\frac{c d e (p+4) \left (a+c x^2\right )^{p+1} (d+e x)^{-2 (p+2)}}{(p+2) (2 p+5) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 745
Rule 837
Rule 807
Rule 727
Rubi steps
\begin{align*} \int (d+e x)^{-6-2 p} \left (a+c x^2\right )^p \, dx &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac{c \int (d+e x)^{-5-2 p} (-d (5+2 p)+3 e x) \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right ) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac{c \int (d+e x)^{-4-2 p} \left (-2 (2+p) \left (3 a e^2-c d^2 (5+2 p)\right )-4 c d e (4+p) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac{c \int (d+e x)^{-3-2 p} \left (2 c d (3+2 p) \left (a e^2 (14+5 p)-c d^2 \left (10+9 p+2 p^2\right )\right )-2 c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) x\right ) \left (a+c x^2\right )^p \, dx}{2 \left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}+\frac{c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}+\frac{\left (c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\right )\right ) \int (d+e x)^{-2-2 p} \left (a+c x^2\right )^p \, dx}{\left (c d^2+a e^2\right )^4 (3+2 p) (5+2 p)}\\ &=-\frac{e (d+e x)^{-5-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right ) (5+2 p)}+\frac{c e \left (3 a e^2 (2+p)-c d^2 \left (18+11 p+2 p^2\right )\right ) (d+e x)^{-3-2 p} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^3 (2+p) (3+2 p) (5+2 p)}+\frac{c^2 d e (3+p) \left (a e^2 (8+5 p)-c d^2 \left (8+7 p+2 p^2\right )\right ) (d+e x)^{-2 (1+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^4 (1+p) (2+p) (3+2 p) (5+2 p)}-\frac{c d e (4+p) (d+e x)^{-2 (2+p)} \left (a+c x^2\right )^{1+p}}{\left (c d^2+a e^2\right )^2 (2+p) (5+2 p)}-\frac{c^2 \left (3 a^2 e^4-6 a c d^2 e^2 (5+2 p)+c^2 d^4 \left (15+16 p+4 p^2\right )\right ) \left (\sqrt{-a}-\sqrt{c} x\right ) \left (-\frac{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (\sqrt{-a}+\sqrt{c} x\right )}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )^{-p} (d+e x)^{-1-2 p} \left (a+c x^2\right )^p \, _2F_1\left (-1-2 p,-p;-2 p;\frac{2 \sqrt{-a} \sqrt{c} (d+e x)}{\left (\sqrt{c} d-\sqrt{-a} e\right ) \left (\sqrt{-a}-\sqrt{c} x\right )}\right )}{\left (\sqrt{c} d+\sqrt{-a} e\right ) \left (c d^2+a e^2\right )^4 (1+2 p) (3+2 p) (5+2 p)}\\ \end{align*}
Mathematica [B] time = 53.7932, size = 5685, normalized size = 10.17 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.599, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{-6-2\,p} \left ( c{x}^{2}+a \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + a\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]